1. Do Now: Pass out calculators. Work on EOC Review Week # 21. Have your agenda out with your assignments written down for a scholar dollar and your current grade.

2. Objective: To Simplify Rational Expressions.

3. Lesson 12.4 Simplify Rational Expressions Rational Expression - Can be written as a fraction but the denominator cannot be 0. *A rational expression is undefined when the denominator is 0. Excluded Value – A number that makes a rational expression undefined.

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. EXAMPLE 1 Find excluded values Find the excluded values, if any, of the expression. d. 7w + 2 8w 2 + w + 5 SOLUTION d. The expression 7w + 2 8w 2 + w + 5 is undefined when 8w 2 + w + 5 = 0. The discriminant is b 2 – 4ac = 1 2 – 4(8)(5) < 0. So, the quadratic equation has no real roots. ANSWER There are no excluded values.

8. 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 2. 2 5y 2 + 2y +3 ANSWER There are no excluded values.

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

10. 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.

11. 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

12. EXAMPLE 2 Simplify the rational expression, if possible. State the excluded values. SOLUTION c. 6m 3 – 12m 3 18m 2 c. 18m 2 6m 3 – 12m 3 = 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

13. 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

14. GUIDED PRACTICE for Example 2 5. 4 a 3 22a 6 2 11a 3 ANSWERThe excluded value is 0. 6. 2c2c c + 5 2c2c ANSWER The excluded value is – 5. 7. 2s 2 + 8s 3s +12 ANSWER 2s2s 3 The excluded value is – 4. 8. 8x8x 8x 3 + 16x 2 ANSWER 1 x 2 + 2x The excluded values are 0 and – 2.

15. 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.

16. 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) (x – 4)(4 + x) = ANSWER The excluded values are – 4 and 4. (x + 4) (x – 3) = –

17. GUIDED PRACTICE for Examples 3 and 4 Simplify the rational expression. State the excluded values. 9. x 2 + 3x + 2 x 2 + 7x + 10 (x + 1) (x + 5) ANSWER The excluded values are – 2 and – 5. 10. y 2 – 64 y 2 – 16y + 64 ANSWER (y + 8) (y – 8) The excluded value is 8 11. 5 + 4z – z 2 z 2 – 3z – 10 ANSWER (z + 1) (z + 2) – The excluded values are 5 and – 2.