Transcript Multiplying Binomials

Multiplying Binomials
Section 8-3 Part 1 & 2
Goals
Goal
• To multiply two binomials
or a binomial by a trinomial.
Rubric
Level 1 – Know the goals.
Level 2 – Fully understand the
goals.
Level 3 – Use the goals to solve
simple problems.
Level 4 – Use the goals to solve
more advanced problems.
Level 5 – Adapts and applies
the goals to different and more
complex problems.
Vocabulary
• None
Multiplying Polynomials
3 Methods for multiplying polynomials
1. Using the Distributive Property
•
Can be used to multiply any two polynomials
2. Using a Table or The Box Method
•
Can be used to multiply any two polynomials
3. Using FOIL
•
Can only be used to multiply two binomials
Method 1: Distributive Property
To multiply a binomial by a binomial, you can
apply the Distributive Property more than once:
(x + 3)(x + 2) = x(x + 2) + 3(x + 2)
= x(x + 2) + 3(x + 2)
Distribute x and 3.
Distribute x and 3 again.
= x(x) + x(2) + 3(x) + 3(2)
Multiply.
= x2 + 2x + 3x + 6
Combine like terms.
= x2 + 5x + 6
Example: Multiply Using
Distributive Property
Multiply.
(s + 4)(s – 2)
(s + 4)(s – 2)
s(s – 2) + 4(s – 2)
s(s) + s(–2) + 4(s) + 4(–2)
Distribute s and 4.
Distribute s and 4 again.
s2 – 2s + 4s – 8
Multiply.
s2 + 2s – 8
Combine like terms.
Your Turn:
Multiply.
(a + 3)(a – 4)
(a + 3)(a – 4)
a(a – 4)+3(a – 4)
Distribute a and 3.
Distribute a and 3 again.
a(a) + a(–4) + 3(a) + 3(–4)
a2 – 4a + 3a – 12
Multiply.
a2 – a – 12
Combine like terms.
Your Turn:
Multiply.
(y + 8)(y – 4)
= y (y – 4) + 8 (y – 4)
= (y2 – 4y) + (8y – 32)
= y2 – 4y + 8y – 32
= y2 + 4y – 32
Method 2: Box Method
Visual model for distributing in polynomial
products, works with any polynomial.
x
Box method
+6
2x
2x2
+ 12x
+1
+ 1x
+6
2 x 2 parts =
2 rows 2 columns
= 2x2 + 12x + 1x + 6
= 2x2 + 13x + 6
Example: Multiply Using
Box Method
Multiply
(x – 3)(4x – 5)
4x
-5
x
4x2
-5x
-3
-12x
+15
= 4x2 – 5x – 12x + 15
= 4x2 – 17x + 15
Your Turn:
Multiply
(3x + 1)(x + 4)
x
+4
3x
3x2
+12x
+1
+x
+4
= 3x2 + 12x + x + 4
= 3x2 + 13x + 4
Your Turn:
Multiply
(2x - 5)(4x + 3)
4x
+3
2x
8x2
+6x
-5
-20x
-15
= 8x2 + 6x - 20x - 15
= 3x2 - 14x - 15
Method 3: FOIL
The product can be simplified using the FOIL
method: multiply the First terms, the Outer terms, the
Inner terms, and the Last terms of the binomials.
First
Last
2
Inner
Outer
Multiplying Polynomials
7-7 Example:
Multiply Using
Multiply
(x + 3)(x + 2)
FOIL
“First Outer Inner Last”, shortcut for distributing,
only works with binomial-binomial products.
F
1. Multiply the First terms. (x + 3)(x + 2)
O
2. Multiply the Outer terms. (x + 3)(x + 2)
I
3. Multiply the Inner terms. (x + 3)(x + 2)
L
4. Multiply the Last terms. (x + 3)(x + 2)
x  x = x2
x  2 = 2x
3  x = 3x
3 2 = 6
(x + 3)(x + 2) = x2 + 2x + 3x + 6 = x2 + 5x + 6
F
Holt Algebra 1
O
I
L
Example: FOIL
F
O
I
L
= z (z) + z (-12) -6 (z) -6 (-12)
= z2
- 12z
– 6z + 72
= z2 - 18z + 72
F
O
I
L
= 5x (2x) + 5x (8) -4 (2x) -4 (8)
= 10x2 + 40x
– 8x – 32
= 10x2 + 32x – 32
Your Turn:
Multiply.
A. (m – 2)(m – 8)
(m – 2)(m – 8)
B. (x + 3)(x + 4)
FOIL
(x + 3)(x + 4)
m2 – 8m – 2m + 16
x2 + 4x + 3x + 12
m2 – 10m +16
x2 + 7x +12
Your Turn:
Multiply.
(x – 3)(x – 1)
(x – 3)(x – 1)
Use the FOIL method.
(x ● x) + (x(–1)) + (–3  x)+ (–3)(–1)
x2 – x – 3x + 3
Multiply.
x2 – 4x + 3
Combine like terms.
Your Turn:
Multiply.
(2a – b2)(a + 4b2)
(2a – b2)(a + 4b2)
Use the FOIL method.
2a(a) + 2a(4b2) – b2(a) + (–b2)(4b2)
2a2 + 8ab2 – ab2 – 4b4
Multiply.
2a2 + 7ab2 – 4b4
Combine like terms.
To multiply polynomials with more than
two terms, you can use the Distributive
Property several times.
Multiply (5x + 3) by (2x2 + 10x – 6):
(5x + 3)(2x2 + 10x – 6) = 5x(2x2 + 10x – 6) + 3(2x2 + 10x – 6)
= 5x(2x2 + 10x – 6) + 3(2x2 + 10x – 6)
= 5x(2x2) + 5x(10x) + 5x(–6) + 3(2x2) + 3(10x) + 3(–6)
= 10x3 + 50x2 – 30x + 6x2 + 30x – 18
= 10x3 + 56x2 – 18
You can also use the Box Method to
multiply polynomials with more than
two terms.
Multiply (5x + 3) by (2x2 + 10x – 6):
2x2
5x
+3
+10x
–6
10x3
50x2
–30x
6x2
30x
–18
Write the product of the
monomials in each row and
column:
To find the product, add all of the terms inside the box by
combining like terms and simplifying if necessary.
10x3 + 6x2 + 50x2 + 30x – 30x – 18
10x3 + 56x2 – 18
Example:
Multiply.
(x – 5)(x2 + 4x – 6)
(x – 5 )(x2 + 4x – 6)
Distribute x and –5.
x(x2 + 4x – 6) – 5(x2 + 4x – 6) Distribute x and −5
again.
x(x2) + x(4x) + x(–6) – 5(x2) – 5(4x) – 5(–6)
x3 + 4x2 – 5x2 – 6x – 20x + 30
Simplify.
x3 – x2 – 26x + 30
Combine like terms.
Your Turn:
Multiply.
(3x + 1)(x3 + 4x2 – 7)
x3
4x2
3x
3x4
12x3
+1
x3
4x2
–7
–21x
–7
Write the product of the
monomials in each row and
column.
Add all terms inside the
rectangle.
3x4 + 12x3 + x3 + 4x2 – 21x – 7
3x4 + 13x3 + 4x2 – 21x – 7
Combine like terms.
Your Turn:
Multiply.
(x + 3)(x2 – 4x + 6)
(x + 3 )(x2 – 4x + 6)
Distribute x and 3.
x(x2 – 4x + 6) + 3(x2 – 4x + 6)
Distribute x and 3 again.
x(x2) + x(–4x) + x(6) +3(x2) +3(–4x) +3(6)
x3 – 4x2 + 3x2 +6x – 12x + 18
Simplify.
x3 – x2 – 6x + 18
Combine like terms.
Your Turn:
= 3a(a2) + 3a(-12a) + 3a(1) + 4(a2) + 4(-12a) + 4(1)
= 3a3 – 36a2 + 3a + 4a2 – 48a + 4
= 3a3 – 32a2 – 45a + 4
Box Method
3 x 3 terms
= 3 by 3 box
2b2
–1
b2
+ 3b
2b4
6b3
-2b2
+ 21b2
-7b
+ 27b
-9
+ 7b + 7b3
+9
Combine like terms
+ 9b2
= 2b4 + 13b3 + 28b2 + 20b – 9
Example: Application
The width of a rectangular prism is 3 feet less
than the height, and the length of the prism is 4
feet more than the height.
Write a polynomial that represents the area of the base of the
prism.
A = lw
A = lw
A = (h + 4)(h – 3)
Write the formula for the area of a
rectangle.
Substitute h – 3 for w and h + 4
for l.
A = h2 + 4h – 3h – 12 Multiply.
A = h2 + h – 12
Combine like terms.
The area is represented by h2 + h – 12.
Your Turn:
The width of a rectangular prism is 3 feet less
than the height, and the length of the prism is
4 feet more than the height.
Find the area of the base when the height is 5 ft.
A = h2 + h – 12
A = h2 + h – 12
Write the formula for the area the base of
the prism.
A = 52 + 5 – 12
Substitute 5 for h.
A = 25 + 5 – 12
Simplify.
A = 18
Combine terms.
The area is 18 square feet.
Assignment
• 8-3 Part 1 Exercises Pg. 498: #4 – 22 even
• 8-3 Part 2 Exercises Pg. 502 – 503: #6 – 30
even