9.4 Multiply and Divide Rational ExpressionsVOCABULARY

Simplified form of a rational expression

VOCABULARYSimplified form of a rational expression

A rational expression in which its numerator and denominator have no common factors (other than ±1 )

SIMPLIFYING RATIONAL EXPRESSIONS

Step 1: Factor numerator and denominator

Step 2: Divide out common factors

Step 3: Simplified form

Example 1Simplify a rational expression(x² + 7x + 10) (x² - 4)Step 1: Factor numerator and denominator(x² + 7x + 10) = (x+2)(x+5) (x² - 4) (x+2)(x-2)Step 2: Divide out common factor(x+2)(x+5)(x+2)(x-2)

Step 3: Simplified Form

(x+5)

(x-2)

CheckpointSimplify the expression

1. x² - 2x – 15

x² + 4x + 3

(x – 5)(x + 3)

(x + 1)(x + 3)

(x – 5)

(x + 1)

Practice!

xx

x

32

42

2

1

12

54

152

2

2

2

2

x

xx

xx

xx

MULTIPLYING RATIONAL EXPRESSIONS

Step 1: Factor numerators and denominators

Step 2: Multiply numerators and denominators

Step 3: Divide out common factors

Step 4: Simplified form

Example 2Multiply rational expressions2x² + 4x x² - 9x + 18x² - 4x – 12 * 2xStep 1: Factor 2x(x + 2) (x – 3)(x – 6)(x + 2)(x – 6) * 2xStep 2: Multiple 2x(x + 2)(x – 3)(x – 6) (x + 2)(x – 6)2x

2x² + 4x x² - 9x + 18x² - 4x – 12 * 2x

Step 3: Divide

2x(x + 2)(x – 3)(x – 6)

(x + 2)(x – 6)2x

Step 4: Simplify

x – 3

CheckpointMultiply the expression6x² + 18x x² - x – 2x² + x – 6 * x² - 7x – 86x(x + 3) (x-2)(x+1)(x+3)(x-2) * (x-8)(x+1)6x(x + 3)(x-2)(x+1)(x+3)(x-2)(x-8)(x+1) 6x (x-8)

Example 3Multiply a rational expression by a polynomial

x – 4 * (x² - x + 1)x³ + 1x – 4 * (x² - x + 1)x³ + 1 1(x-4)(x² - x + 1)(x+1)( x² - x + 1)X – 4 X + 1

Dividing Rational ExpressionsStep 1: Multiply by reciprocal

Step 2: Factor

Step 3: Divide

Step 4: Simplify

Example 4Divide rational expressions 3 ● 8x² - 8x x + 7 ● x² + 6x – 7Step 1: Multiply by reciprocal 3 x² + 6x – 7x+7 * 8x² - 8x Step 2: Factor 3 (x+7)(x-1)(x+7)(8x)(x-1)

3 ● 8x² - 8x x + 7 ● x² + 6x – 7 cont.

Step 3: Divide

3 (x+7)(x-1)

(x+7)(8x)(x-1)

Step 4: Simplify

3__

8x

CheckpointDivide the expression3. x – 5 ● 2x² - 11x + 5 9x² - 18x ● 2x² - 5x + 2 (x – 5) (2x² - 5x + 2) (9x² - 18x)(2x² - 11x + 5)(x – 5)(2x-1)(x-2)9x(x–2)(2x-1)(x-5) 1_9x

A farmer wants to fence in a square field

that measures 2x on each side.

Write a simplified rational

expression for the ratio of the

field’s perimeter to its area.

From 1987 to 1996 the total acres of farmland L ( in millions) and the total number of farms F ( in hundreds of thousands ) in the United States can be modeled by:

t represents the number of years since 1987. Write a model for the average number of acres per

farm as a function of the year. What was the average number of acres per farm in

1993?

105.

20.2101.

10482.

9993.43

2

2

t

tF

t

tL

Warm-up

12

1

36

632

2

x

x

xx

xx

n

n

n

nn

10

2424

10

30273 2

)42(28182

502

nnn

n344

2

5

5

5

25

xx

x

x

x