1. 8.4 Multiply and Divide Rational Expressions
VOCABULARY
Simplified form of a rational expression

2. VOCABULARY Simplified form of a rational expression
A rational expression in which its numerator and denominator have no common factors (other than ±1 )

3. SIMPLIFYING RATIONAL EXPRESSIONS
Step 1: Factor numerator and denominator
Step 2: Divide out common factors
Step 3: Simplified form

4. 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)

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

6. 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

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

8. 2x² + 4x x² - 9x + 18 x² - 4x – 12 * 2x Step 3: Divide
Step 4: Simplify
x – 3

9. Checkpoint Multiply the expression
6x² + 18x x² - x – 2
x² + x – * 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)

10. Example 3 Multiply a rational expression by a polynomial
x – 4 * (x² - x + 1)
x³ + 1 x³
(x-4)(x² - x + 1)
(x+1)( x² - x + 1)
X – 4
X + 1

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

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

13. 3 ● 8x² - 8x x + 7 ● x² + 6x – 7 cont. Step 3: Divide 3 (x+7)(x-1)
Step 4: Simplify
3__ 8x

14. Checkpoint Divide the expression
3. x – ● x² - 11x + 5
9x² - 18x ● x² - 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

15. 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.

16. 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?