Ch. 5.5 power point
Download
Report
Transcript Ch. 5.5 power point
Chapter 5
Section 5
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
5.5
Multiplying Polynomials
1
Multiply a monomial and a polynomial.
2
Multiply two polynomials.
3
Multiply binomials by the FOIL method.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 1
Multiply a monomial and a
polynomial.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5.5 - 3
Multiply a monomial and a polynomial.
As shown in Section 5.1, we find the product of two
monomials by using the rules for exponents and the
commutative and associative properties. For example
8m6 9n6 8 9 m6 n 6 72m6 n 6 .
To find the product of a monomial and a polynomial with
more than one term we use the distributive property and
multiplication of monomials.
Do not confuse addition of terms with multiplication of terms.
For instance,
7q5 2q5 9q5 , but
7q 2q 7 2q
5
5
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
55
14q10 .
Slide 5.5 - 4
EXAMPLE 1
Multiplying Monomials and
Polynomials
Find the product.
2 x 4 3x 2 2 x 5
Solution:
2 x 3x 2 x 2 x 2 x 5
4
2
6 x 4 x 10 x
6
5
4
4
4
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5.5 - 5
Objective 2
Multiply two polynomials.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5.5 - 6
Multiply two polynomials.
We can use the distributive property repeatedly to find the
product of any two polynomials. For example, to find the
product of the polynomials x2 + 3x +5 and x − 4, think of x − 4 as
a single quantity and use the distributive property as follows.
x
2
3x 5 x 4 x 2 x 4 3x x 4 5 x 4
Now use the distributive property three more times to find
x2(x − 4), 3x(x − 4), and 5(x − 4).
x2 x x2 4 3x x 3x 4 5 x 5 4
x3 4 x 2 3x 2 12 x 5 x 20
x3 x2 7 x 20
This example suggests the following rule.
To multiply polynomials, multiply each term of the second
polynomial by each term of the first polynomial and add the
products.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5.5 - 7
EXAMPLE 2
Multiplying Two Polynomials
Multiply (m3 − 2m + 1)·(2m2 + 4m + 3).
Solution:
m3 2m2 m3 4m m3 3 2m 2m2 2m 4m
2m 3 1 2m2 1 4m 1 3
2m5 4m 4 3m3 4m3 8m 2 6m 2m 2 4m 3
2m5 4m4 m3 6m2 2m 3
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5.5 - 8
EXAMPLE 3
Multiplying Polynomials Vertically
Multiply.
3x 2 4 x 5
x4
Solution:
12 x2 16 x 20
3x3 4 x 2 5 x
3x3 16 x2 11x 20
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5.5 - 9
EXAMPLE 4
Multiplying Polynomials with
Fractional Coefficients Vertically
Multiply.
5x3 10 x 2 20
1 2 2
x
5
5
Solution:
2 x3 4 x 2 8
x5 2 x 4 0 x3 4 x 2
x5 2 x 4 2 x3 8
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5.5 - 10
EXAMPLE 4A
Multiplying Polynomials with a
Rectangle Model
Use the rectangle method to find the product
4x 3 x 2.
Solution:
4x 2
3x
8x
6
4 x2 11x 6
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5.5 - 11
Objective 3
Multiply binomials by the FOIL
method.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5.5 - 12
Multiply binomials by the FOIL method.
In algebra, many times the polynomials to be multiplied are
binomials. For these products, the FOIL method reduces the
rectangle method to a systematic approach without the rectangle.
A summary of the steps in the FOIL method follows.
Step 1: Multiply the two First terms of the binomials to get the
first term of the answer.
Step 2: Find the Outer product and Inner product and add them
(when possible) to get the middle term of the answer.
Step 3: Multiply the two Last terms of the binomials to get the
last term of the answer.
L 15
F x2
x 3 x 5
O 5x
I 3x
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5.5 - 13
EXAMPLE 5
Using the FOIL Method
Find the product by the FOIL method.
F x
L 12
2
x 2 x 6
O 6x I 2x
Solution:
x2 6 x 2 x 8
x 8x 12
2
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5.5 - 14
EXAMPLE 6
Using the FOIL Method
Multiply 5x 6 2 y 3 .
F 10xy
L 18
5x 6 2 y 3
O 15x
I 12 y
Solution:
10 xy 15x 12 y 18
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5.5 - 15
EXAMPLE 7
Using the FOIL Method
Find each product.
4 y x 2 y 3x
Solution:
8 y 12xy 2xy 3x
8 y 2 14xy 3x2
3
3x x 2 2x 1
2
2
3 x 3 2 x 2 1x 4 x 2
3x3 2 x 2 3x 2
6x 9x 6x
5
4
3
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5.5 - 16