Beginning Algebra, 11th Edition

Multiplying Two Polynomials

Multiply

= 4 m^5 - 2 m^4 + 24 m^3 - 10 m^2 + 20 m Combine like terms. NOW TRY

= 4 m^5 - 2 m^4 + 4 m^3 + 20 m^3 - 10 m^2 + 20 m

= m^214 m^32 + m^21 - 2 m^22 + m^214 m 2 + 514 m^32 + 51 - 2 m^22 + 514 m 2

1 m^2 + 5214 m^3 - 2 m^2 + 4 m 2

1 m^2 + 5214 m^3 - 2 m^2 + 4 m 2.

NOW TRY EXAMPLE 2

EXERCISE 2
Multiply.

1 x^2 - 4212 x^2 - 5 x+ 32

NOW TRY ANSWERS

2. 2 x^4 - 5 x^3 - 5 x^2 + 20 x- 12

NOW TRY
EXERCISE 3
Multiply.

2 t- 6

5 t^2 - 7 t+ 4

NOW TRY
EXERCISE 4
Find the product of

9 x^3 - 12 x^2 + 3 and.^13 x^2 - 32

3. 10 t^3 - 44 t^2 + 50 t- 24


4. 3 x^5 - 4 x^4 - 6 x^3 + 9 x^2 - 2

330 CHAPTER 5 Exponents and Polynomials

3 x^4 + 11 x^3 + 22 x^2 + 23 x+ 5 Product NOW TRY

3 x^4 + 6 x^3 + 12 x^2 + 3 x 3 x 1 x^3 + 2 x^2 + 4 x+ 12

5 x^3 + 10 x^2 + 20 x+ 5

3 x+ 5

x^3 + 2 x^2 + 4 x+ 1

Multiplying Polynomials

To multiply two polynomials, multiply each term of the second polynomial

by each term of the first polynomial and add the products.

Place like terms in

columns so they

can be added.

Write the polynomials

vertically

51 x^3 + 2 x^2 + 4 x+ 12

This process is similar to

multiplication of whole numbers.

Multiplying Polynomials Vertically

Multiply vertically.

Begin by multiplying each of the terms in the top row by 5.

Now multiply each term in the top row by 3x. Then add like terms.

5 x^3 + 10 x^2 + 20 x+ 5

3 x+ 5

x^3 + 2 x^2 + 4 x+ 1

3 x+ 5

x^3 + 2 x^2 + 4 x+ 1

1 x^3 + 2 x^2 + 4 x+ 1213 x+ 52

EXAMPLE 3

Multiply each term of the second

polynomial by each term of the first.

Multiplying Polynomials with Fractional Coefficients Vertically

Find the product of and

Terms of top row are multiplied by.

Terms of top row are multiplied by.

Add. NOW TRY

We can use a rectangle to model polynomial multiplication. For example, to find

label a rectangle with each term as shown next on the left. Then put the product of

each pair of monomials in the appropriate box, as shown on the right.

22

113 x 2

2 x 2 x 6 x^24 x

3 x 3 x

12 x+ 1213 x+ 22 ,

2 m^5 - m^4 + 12 m^3 - 5 m^2 + 10 m

1

2 m

2 m^5 - m^4 + 2 m^32

5

10 m 2

(^3) - 5 m (^2) + 10 m
1

2 m

(^2) +^5
2

4 m^3 - 2 m^2 + 4 m

1

2 m

(^2) +^5

4 m 2.

(^3) - 2 m (^2) + 4 m

EXAMPLE 4

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