1. 3.8 Warm Up 1. (18x³ - 24x² + 12x) ∕ 6x 2. (5x² + 7x – 6) ∕ (x + 2) 3. (9x² + 5x – 6) ∕ (x + 1)

2. 3.8 Simplifying Rational Expressions

3. Vocabulary Rational Expression: ratio of 2 polynomials Excluded Value: once simplified, the value that would make the denominator 0 Simplest Form: when numerator & denominator have no common factors

4. EXAMPLE 1 Find excluded values Find the excluded values, if any, of the expression. a. x + 8 10x SOLUTION a. The expression x + 8 is undefined when 10x = 0, or x = 0. 10x ANSWER The excluded value is 0.

5. EXAMPLE 1 Find excluded values Find the excluded values, if any, of the expression. b. 2y + 14 5 SOLUTION The expression 5 is undefined when 2y + 14 = 0, or x = – 7. 2y + 14 ANSWER The excluded value is – 7.

6. EXAMPLE 1 Find excluded values Find the excluded values, if any, of the expression. c. v 2 – 9 4v4v SOLUTION c.c. The expression 4v is undefined when v 2 – 9 = 0, or (v + 3)(v – 3) = 0. The solutions of the equation are – 3 and 3. v2 – 9v2 – 9 The excluded values are – 3 and 3. ANSWER

7. GUIDED PRACTICE for Example 1 Find the excluded values, if any, of the expression. x + 2 3x – 5 1. ANSWER The excluded value is. 5 3 3. ANSWER The excluded value is and 4. 3 2 – n – 6 2n 2 – 5n – 12 2. 2m m 2 – 4 ANSWER The excluded value is 2, and 2. –

8. EXAMPLE 2 Simplify expressions by dividing out monomials Simplify the rational expression, if possible. State the excluded values. a. r 2r SOLUTION Divide out common factor. a. r 2r2r = r 2r2r = 1 2 Simplify. ANSWER The excluded value is 0.

9. EXAMPLE 2 Simplify the rational expression, if possible. State the excluded values. b. 5x5x 5(x + 2) SOLUTION b. 5x5x 5(x + 2) = 5x Divide out common factor. Simplify. = x (x + 2) ANSWER The excluded value is – 2. Simplify expressions by dividing out monomials

10. EXAMPLE 2 Simplify the rational expression, if possible. State the excluded values. SOLUTION c. 6m 3 – 12m 2 18m 2 c. 18m 2 6m 3 – 12m 2 = 6m 2 (m – 2) 63m2m2 Factor numerator and denominator. = 6m 2 (m – 2) 6 3 m2m2 Divide out common factors. = m – 2 3 Simplify. ANSWER The excluded value is 0. Simplify expressions by dividing out monomials

11. EXAMPLE 2 Simplify the rational expression, if possible. State the excluded values. SOLUTION d. y 7 – y d. The expression y 7 – y is already in simplest form. ANSWER The excluded value is 7. Simplify expressions by dividing out monomials

12. GUIDED PRACTICE for Example 2 5. 4 a 3 22a 6 6. 2c2c c + 5 7. 2s 2 + 8s 3s +12 8. 8x8x 8x 3 + 16x 2

13. EXAMPLE 3 Simplify an expression by dividing out binomials Simplify x 2 – 3x – 10 x 2 + 6x + 8. State the excluded values. SOLUTION x 2 – 3x – 10 x 2 + 6x + 8 = (x – 5)(x + 2) (x + 4)(x + 2) Factor numerator and denominator. (x – 5)(x + 2) (x + 4)(x + 2) = = x – 5 x + 4 Divide out common factor. Simplify. ANSWER The excluded values are – 4 and – 2.

14. EXAMPLE 4 Recognize opposites Simplify x 2 – 7x + 12 16 – x 2. State the excluded values. SOLUTION x 2 – 7x + 12 16 – x 2 Factor numerator and denominator. (x – 3)(x – 4) – (x – 4)(4 + x) = Rewrite 4 – x as – ( x – 4). Simplify. – (x – 4)(4+ x) (x – 3)(x – 4) = Divide out common factor. –(4 + x) (x – 3) = (x – 3)(x – 4) (4 – x)(4 + x) = ANSWER The excluded values are – 4 and 4. (x + 4) (x – 3) = –

15. GUIDED PRACTICE for Examples 3 and 4 Simplify the rational expression. State the excluded values. 9. x 2 + 3x + 2 x 2 + 7x + 10 10. y 2 – 64 y 2 – 16y + 64 11. 5 + 4z – z 2 z 2 – 3z – 10