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Chapter Chapter 55Section Section 55
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Multiplying Polynomials
11
33
22
5.55.55.55.5Multiply a monomial and a polynomial.
Multiply two polynomials.
Multiply binomials by the FOIL method.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 11
Multiply a monomial and a polynomial.
Slide 5.5 - 3
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
To find the product of a monomial and a polynomial with more than one term we use the distributive property and multiplication of monomials.
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
Slide 5.5 - 4
6 6 6 6 6 68 9 8 9 .72m n m n m n
Do not confuse addition of terms with multiplication of terms.
For instance,
, but5 5 57 2 9q q q 5 5 5 5 107 2 7 2 14 .q q q q
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 1 Multiplying Monomials and Polynomials
Solution:
Slide 5.5 - 5
Find the product.
4 22 3 2 5x x x
424 43 2 52 2 2x x xx x
6 5 46 4 10x x x
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 22
Multiply two polynomials.
Slide 5.5 - 6
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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.
Slide 5.5 - 7
2 24 4 45 5 43 3x x xx x xx x
Now use the distributive property three more times to find x2(x − 4), 3x(x − 4), and 5(x − 4).
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.
2 2 3 3 5 54 4 4x xx x x x x
3 2 24 3 12 5 20x x x x x 3 2 7 20x x x
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 2 Multiplying Two Polynomials
Slide 5.5 - 8
Multiply (m3 − 2m + 1)·(2m2 + 4m + 3).
Solution:
3 2 3 3 2
2
2 4 3 2 2 2 4
2 3 1 2 1 4 1 3
m m m m m m m m m
m m m
5 4 3 3 2 22 4 3 4 8 6 2 4 3m m m m m m m m
5 4 3 22 4 6 2 3m m m m m
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 3 Multiplying Polynomials Vertically
Slide 5.5 - 9
Multiply.23 4 5x x
4x 212 16 20x x
3 23 4 5x x x 3 23 16 11 20x x x
Solution:
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 4 Multiplying Polynomials with Fractional Coefficients Vertically
Slide 5.5 - 10
Multiply.3 25 10 20x x
21 2
5 5x
3 22 4 8x x 5 4 3 22 0 4x x x x
5 4 32 2 8x x x
Solution:
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 4A Multiplying Polynomials with a Rectangle Model
Slide 5.5 - 11
Use the rectangle method to find the product 4 3 2 .x x
24x 8x
3x 6
24 11 6x x
Solution:
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 33
Multiply binomials by the FOIL method.
Slide 5.5 - 12
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Multiply binomials by the FOIL method.
Slide 5.5 - 13
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.
3 5x x
2F x L 15
O 5x I 3x
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 5 Using the FOIL Method
Slide 5.5 - 14
Find the product by the FOIL method.
2 6x x
2 6 2 8x x x 2 8 12x x
Solution:
2F x
O 6x I 2x
L 12
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 6 Using the FOIL Method
Slide 5.5 - 15
Multiply 5 6 2 3 .x y
10 15 12 18xy x y Solution:
5 6 2 3x y
F 10xy L 18
O 15x I 12y
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 7 Using the FOIL Method
Slide 5.5 - 16
Find each product.
4 2 3y x y x
2 28 12 2 3y xy xy x Solution:
33 2 2 1x x x
2 28 14 3y xy x
3 23 2 1 4 2x x x x
3 23 2 3 2x x x 5 4 36 9 6x x x