3.9 Multiplying & Dividing Rational

Expressionsp. 165-168

Multiplying Factor numerator and denominator of each

expression Combine both numerators on top Combine both denominators on bottom Look to cancel any common factors Simplify

Multiply rational expressions involving monomialsEXAMPLE 1

Find the product 2x2

3x

6x2

12x3

Multiply numerators and

denominators.

Product of powers property

Factor and divide out common factors.

Simplify.13=

12 x4

3 12 x4 =

12x4

36x4=

2x2

3x

6x2

12x3=

(2x2)(6x2)

(3x)(12x3)

GUIDED PRACTICE for Example 1

Find the product

2y3

5y

15y3

8y5

1.

34ANSWER

2.7z2

4z3

z3

14z

ANSWERz8

EXAMPLE 2 Multiply rational expressions involving polynomials

Find the product 3x2 + 3x4x2 – 24x + 36

x2 – 4x + 3x2 – x

Multiply numerators and denominators.

3x2 + 3x4x2 – 24x + 36

x2 – 4x + 3x2 – x

=(3x2 + 3x) (x2 – 4x + 3)(4x2 – 24x + 36)(x2 – x)

Factor and divide out common factors.

= 3(x + 1)4(x – 3)

Simplify.

= 3x(x + 1)(x – 3)(x – 1)4x(x – 3)(x – 3)(x – 1)

EXAMPLE 2 Multiply rational expressions involving polynomials

CHECK

Check your simplification using a graphing calculator.

The graphs coincide. So, the expressions are equivalent for all values of x other than the excluded values (0, 1, and 3).

Graph y1 = 3x2 + 3x4x2 – 24x + 36

x2 – 4x + 3x2 – x

and y2 =3(x + 1)4(x – 3)

.

EXAMPLE 3 Multiply a rational expression by a polynomial

Find the product 5x

x2 + 5x + 6(x + 3).

5x

x2 + 5x + 6(x + 3).

5x

x2 + 5x + 6

(x + 3)=

1

Rewrite polynomial as a fraction.

= 5x(x + 3)x2 + 5x + 6

Multiply numerators and denominators.

=5x(x + 3)

(x + 3)(x + 2)

Factor and divide out common factor.

Simplify.=5x

x + 2

GUIDED PRACTICE for Examples 2 and 3

3. Find the product x2 + x – 2 x2 + x

2x2 + 2x5x2 –15x +10

2(x + 2)5(x – 2)

ANSWER

Find the product2w2

w2 – 7w + 12(w – 4).4.

2w2

w – 3ANSWER

Dividing Multiply by the multiplicative inverse

(reciprocal) of the second expression Factor & write as one rational expression Cancel/divide out common factors Simplify

EXAMPLE 4 Divide rational expressions involving polynomials

Find the quotient .7x2 – 7xx2 + 2x – 3

x + 1x2 – 7x – 8

7x2 – 7xx2 + 2x – 3

x + 1x2 – 7x – 8

7x2 – 7xx2 + 2x – 3 x + 1

= x2 – 7x – 8 Multiply by multiplicative inverse.

= (7x2 – 7x) (x2 – 7x – 8)(x + 1)(x2 + 2x – 3)

Multiply numerators and denominators.

= 7x(x – 1)(x – 8)(x + 1)(x + 3)(x – 1)(x + 1)

Factor and divide out common factors.

=7x(x – 8)

x + 3Simplify.

EXAMPLE 5 Divide a rational expression by a polynomial

Find the quotient 2x2 + 16x + 243x2

(x +6).

2x2 + 16x + 243x2

(x +6)=

2x2 + 16x + 243x2

(x +6).= 1

Rewrite polynomial as fraction.

2x2 + 16x + 243x2=

1(x +6).

Multiply by multiplicative inverse.

2x2 + 16x + 243x2= (x +6).

Multiply numerators and denominators.

=2(x + 2)(x + 6)

3x2(x + 6)Factor and divide out common factor.

=2(x + 2)

3x2 Simplify.

GUIDED PRACTICE for Examples 4 and 5

5. m2 – 4

2m2 + 4m

6m – 3m2

4m + 44

2 (m +11)

3m2

–ANSWER

6. n2 – 6n + 912n

n – 3

ANSWER12n

n – 3

Find the quotient.

EXAMPLE 6 Solve a multi-step problem

A = 13,000 + 3700x 1 – 0.015x

and T = 1800 + 860x1 – 0.016x

The amount A (in millions of dollars) spent on all advertising and the amount T (in millions of dollars) spent on television advertising in the United States during the period 1970–2003 can be modeled by

where x is the number of years since 1970. Write a model that gives the percent p (in decimal form) of the amount spent on all advertising that was spent on television advertising. Then approximate the percent spent on television advertising in 2003.

ADVERTISING

EXAMPLE 6 Solve a multi-step problem

STEP 1

SOLUTION

Write a verbal model. Then write an equation.

P = T A

p = T A Write equation.

1800 +860x1– 0.016

13,000 + 3700x1– 0.015x x

=

STEP 2 Find the quotient.

Substitute for T and for A.

EXAMPLE 6 Solve a multi-step problem

1800 +860x1–0.016x 13,000 3700x+

1– 0.015x=

(1800 + 860x)(1 –0.015x)

(1– 0.016x)(13,000 + 3700x)=

Multiply by multiplicative inverse.

Multiply numerators and denominators.

Factor and divide out common factor.

(90 + 43x)(1 – 0.015x)

(1 – 0.016x)(650 + 185x)= Simplify.

20 (90 + 43x) (1 – 0.015x)

(1 – 0.016x)(20)(650 + 185x)=

EXAMPLE 6 Solve a multi-step problem

STEP 3 Approximate the percent spent on television advertising in 2003.Because 2003 – 1970 = 33, x = 33. Substitute 33 for x in the model and use a calculator to evaluate.

ANSWER About 24% of the amount spent on all advertising was spent on television advertising in 2003.

(90 + 43 33)(1 – 0.015 33)

(1 – 0.016 33)(650 + 185 33)P = 0.239

GUIDED PRACTICE for Example 6

7. In Example 6, find the values of T and of A separately when x = 33. Then divide the value of T by the value of A. Compare your answer with the answer in Step 3 above.

ANSWER

About $63,941 million

About $267,525 million

About 0.239

The answers are the same.