Section 5.3 Special Products Copyright © 2013, 2009, 2006 Pearson Education, Inc. Objective #1 Use FOIL in polynomial multiplication. Copyright © 2013, 2009, 2006 Pearson Education,
Download ReportTranscript Section 5.3 Special Products Copyright © 2013, 2009, 2006 Pearson Education, Inc. Objective #1 Use FOIL in polynomial multiplication. Copyright © 2013, 2009, 2006 Pearson Education,
Slide 1
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 2
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 3
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 4
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 5
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 6
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 7
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 8
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 9
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 10
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 11
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 12
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 13
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 14
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 15
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 16
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 17
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 18
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 19
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 20
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 21
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 22
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 23
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 24
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 25
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 26
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 27
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 28
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 29
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 30
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 31
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 32
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 2
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 3
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 4
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 5
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 6
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 7
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 8
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 9
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 10
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 11
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 12
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 13
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 14
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 15
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 16
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 17
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 18
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 19
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 20
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 21
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 22
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 23
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 24
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 25
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 26
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 27
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 28
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 29
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 30
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 31
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
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12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32
Slide 32
Section 5.3
Special
Products
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
Objective #1
Use FOIL in polynomial multiplication.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Special Products
In this section we will use the distributive property to develop patterns that will
allow us to multiply some special binomials quickly.
We will find the product of two binomials using a method called FOIL.
We will learn a formula for finding the square of a binomial sum. We will also
learn formula for finding the product of the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
Multiplying Polynomials - FOIL
Using the FOIL Method to Multiply Binomials
last
first
F
O
I
L
ax bcx d ax cx ax d b cx b d
inside
outside
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Product of
Last terms
4
Multiplying Polynomials - FOIL
EXAMPLE
Multiply 4 x 35x 1 .
SOLUTION
last
F
O
I
L
first
4x 35x 1 4 x 5x 4 x 1 3 5x 3 1
inside
outside
20x 2 4 x 15x 3
Multiply
20x 2 19x 3
Combine like terms
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
FOIL Method
EXAMPLE
Multiply: (5x + 2)(x + 7)
F
O
I
L
(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7
Product
of the
first
terms
Product
of the
outside
terms
Product
of the
inside
terms
Product
of the
last
terms
=5x2 + 35x + 2x +14
=5x2 + 37x +14
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Objective #1: Example
1a. Multiply: ( x 5)( x 6)
F
O
I
L
( x 5)( x 6) x x 6 x 5 x 5 6
x 2 6 x 5 x 30
x 2 11x 30
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
1b. Multiply: (7 x 5)(4 x 3)
F
O
I
L
(7 x 5)(4 x 3) 7 x 4 x 7 x( 3) 5 4 x 5( 3)
28 x 2 21x 20 x 15
28 x 2 x 15
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #2
Multiply the sum and difference of two terms.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Multiplying the Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2
The product of the sum and the difference of
the same two terms is the square of the first
term minus the square of the second term.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
Objective #2: Example
2a. Multiply: (7 y 8)(7 y 8)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
second term
squared
(7 y 8)(7 y 8) (7 y ) 2
82
49 y 2 64
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Objective #2: Example
2b. Multiply: (2a3 3)(2a3 3)
Since this product is of the form ( A B)( A B) ,
use the special–product formula ( A B)( A B) A2 B2 .
first term
squared
(2a3 3)(2a3 3) (2a3 ) 2
second term
squared
32
4a 6 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #3
Find the square of a binomial sum.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
The Square of a Binomial Sum
(A + B)2 = A2 + 2AB + B2
The square of a binomial sum is the first
term squared plus two times the product
of the terms plus the last term squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 4x y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
4x y 2
A2
4x 2
2
+
2 AB
2 Product
B2
+
Last
Term
+
y2
of theT erms
+
2 4x y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
16x 2 8xy y 2
20
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Objective #3: Example
3a. Multiply: ( x 10)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 10) 2
x2
2 product
of the terms
last term
squared
2 10 x 102
x 2 20 x 100
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
23
Objective #3: Example
3b. Multiply: (5 x 4)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(5 x 4) 2 (5 x) 2 2 20 x
last term
squared
42
25 x 2 40 x 16
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
24
Objective #4
Find the square of a binomial difference.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
25
The Square of a Binomial Difference
(A – B)2 = A2 – 2AB + B2
The square of a binomial difference is the
first term squared minus two times the
product of the terms plus the last term
squared.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
26
Multiplying Polynomials – Special Formulas
EXAMPLE
Multiply 3x 4 y 2 .
SOLUTION
Use the special-product formula shown.
A B2
First
Term
3x 4 y 2
A2
3x 2
2
–
2 AB
2 Product
B2
+
Last
Term
+
4y 2
of theT erms
–
2 3x 4 y
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
= Product
9x 2 24xy 16y 2
27
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
28
Objective #4: Example
4a. Multiply: ( x 9)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
( x 9) 2
x2
last term
squared
2 product
of the terms
2 9x
92
x 2 18 x 81
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
29
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
30
Objective #4: Example
4b. Multiply: (7 x 3)2
Use the special-product formula ( A B)2 A2 2 AB B2 .
first term
squared
2 product
of the terms
(7 x 3) 2 (7 x) 2 2 21x
last term
squared
32
49 x 2 42 x 9
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
31
Multiplying Polynomials – Special Formulas
The Square of a Binomial Sum
A B2 A2 2 AB B2
The Square of a Binomial Difference
A B2 A2 2 AB B2
The Product of the Sum and
Difference of Two Terms
A BA B A2 B2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
32