Chapter 5 Section 5 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

Chapter Chapter 55Section Section 55

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.


Objective 11

Multiply a monomial and a polynomial.

Slide 5.5 - 3


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


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


Objective 22

Multiply two polynomials.

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.

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


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


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:


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:


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:


Objective 33


Slide 5.5 - 12



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


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



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



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

Chapter 5 Section 5 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

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Transcript of Chapter 5 Section 5 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.