Intermediate Algebra (11th edition)

298 CHAPTER 5 Exponents, Polynomials, and Polynomial Functions

(d)

Square twice.

Distributive property

Distributive property again

Combine like terms. NOW TRY

OBJECTIVE 6 Multiply polynomial functions. In Section 5.3,we added and

subtracted functions. Functions can also be multiplied.

= 16 a^4 + 32 a^3 b+ 24 a^2 b^2 + 8 ab^3 +b^4

+ 4 a^2 b^2 + 4 ab^3 + b^4

= 16 a^4 + 16 a^3 b+ 4 a^2 b^2 + 16 a^3 b+ 16 a^2 b^2 + 4 ab^3

+b^214 a^2 + 4 ab+b^22

= 4 a^214 a^2 + 4 ab+ b^22 + 4 ab 14 a^2 + 4 ab+b^22

= 14 a^2 + 4 ab+ b^2214 a^2 + 4 ab+ b^222 a+b

= 12 a+b 22 12 a+b 22

12 a+b 24

CAUTION Write the product as not which indi-

cates the composition of functions ƒ and g. (See Section 5.3.)

ƒ 1 x 2 #g 1 x 2 1 ƒg 21 x 2 , ƒ 1 g 1 x 22 ,

Multiplying Functions

If and define functions, then

Product function

The domain of the product function is the intersection of the domains of

and g 1 x 2.

ƒ 1 x 2

1 ƒg 21 x 2 ƒ 1 x 2 #g 1 x 2.

ƒ 1 x 2 g 1 x 2

Multiplying Polynomial Functions

For and find and

Use the definition. Substitute. FOIL Combine like terms.

Let in

Add and subtract. Confirm that ƒ 1 - 12 #g 1 - 12 is equal to 1 ƒg 21 - 12. NOW TRY

= 1

=- 6 + 11 - 4

= 61 - 123 + 111 - 122 + 41 - 12 x=- 1 1 ƒg 21 x 2.

1 ƒg 21 - 12

= 6 x^3 + 11 x^2 + 4 x

= 6 x^3 + 3 x^2 + 8 x^2 + 4 x

= 13 x+ 4212 x^2 +x 2

=ƒ 1 x 2 #g 1 x 2

1 ƒg 21 x 2

ƒ 1 x 2 = 3 x+ 4 g 1 x 2 = 2 x^2 +x, 1 ƒg 21 x 2 1 ƒg 21 - 12.

EXAMPLE 8

Be careful with signs.

NOW TRY
EXERCISE 7
Find each product.

(a)

(b) 1 y- 324

314 x-y 2 + 24314 x-y 2 - 24

NOW TRY
EXERCISE 8
For
and
find and. 1 ƒg 21 x 2 1 ƒg 21 - 22

g 1 x 2 = 8 x+7,

ƒ 1 x 2 = 3 x^2 - 1

NOW TRY ANSWERS

(a)
(b)

24 x^3 + 21 x^2 - 8 x-7; - 99

108 y+ 81

y^4 - 12 y^3 + 54 y^2 -

16 x^2 - 8 xy+y^2 - 4