9.4 Multiply and Divide Rational Expressions VOCABULARY Simplified form of a rational expression

VOCABULARY Simplified 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 1 Simplify 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)

Checkpoint Simplify the expression 1. x² - 2x – 15 x² + 4x + 3 (x – 5)(x + 3) (x + 1)(x + 3) (x – 5) (x + 1)

Practice!

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

2x² + 4x x² - 9x + 18 x² - 4x – 12 * 2x Step 3: Divide 2x(x + 2)(x – 3)(x – 6) (x + 2)(x – 6)2x Step 4: Simplify x – 3

Checkpoint Multiply the expression 6x² + 18x x² - x – 2 x² + x – 6 * x² - 7x – 8 6x(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 3 Multiply a rational expression by a polynomial x – 4 * (x² - x + 1) x³ + 1 x – 4 * (x² - x + 1) x³ (x-4)(x² - x + 1) (x+1)( x² - x + 1) X – 4 X + 1

Dividing Rational Expressions Step 1: Multiply by reciprocal Step 2: Factor Step 3: Divide Step 4: Simplify

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

Checkpoint Divide the expression 3. 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  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?

Warm-up