1. 12.5 Warm Up Warm Up Lesson Quiz Lesson Quiz Lesson Presentation Lesson Presentation Multiply and Divide Rational Expressions

2. 12.5 Warm-Up 1.x 2 + 3x – 18 2.2x 2 – 4x – 16 ANSWER (x – 3)(x + 6) ANSWER 2(x – 4)(x + 2) Factor the polynomial. 3.2x 2 + 8x – 10 ANSWER 2(x + 5)(x – 1)

3. 12.5 Warm-Up 4. A golf clinic charges a $30 enrollment fee and $5 per session. What is the average cost per session if you attend 8 sessions? ANSWER $8.75

4. 12.5 Example 1 Find the product. 2x22x2 3x 6x 2 12x 3 Multiply numerators and denominators. Product of powers property Factor and divide out common factors. Simplify. 1 3 = 12 x 4 3 12 x 4 = 12x 4 36x 4 = 2x22x2 3x 6x26x2 12x 3 = (2x 2 )(6x 2 ) (3x)(12x 3 )

5. 12.5 Guided Practice Find the product 2y32y3 5y 15y 3 8y 5 1. 3 4 ANSWER 2. 7z27z2 4z 3 z 3 14z ANSWER z 8

6. 12.5 Example 2 Find the product 3x 2 + 3x 4x 2 – 24x + 36 x 2 – 4x + 3 x 2 – x Multiply numerators and denominators. 3x 2 + 3x 4x 2 – 24x + 36 x 2 – 4x + 3 x 2 – x = (3x 2 + 3x) (x 2 – 4x + 3) (4x 2 – 24x + 36)(x 2 – x) Factor and divide out common factors. = 3(x + 1) 4(x – 3) Simplify. = 3x(x + 1)(x – 3)(x – 1) 4x(x – 3)(x – 3)(x – 1)

7. 12.5 Example 2 CHECK Check your simplification using a graphing calculator. The graphs coincide. So, the expressions are equivalent for all values of x other than the excluded values (0, 1, and 3). Graph y 1 = 3x 2 + 3x 4x 2 – 24x + 36 x 2 – 4x + 3 x 2 – x and y 2 = 3(x + 1) 4(x – 3).

8. 12.5 Guided Practice 3. Find the product x 2 + x – 2 x 2 + x 2x 2 + 2x 5x 2 –15x +10 2(x + 1) 5(x – 2) ANSWER

9. 12.5 Example 3 Find the product 5x5x x 2 + 5x + 6 (x + 3). 5x5x x 2 + 5x + 6 (x + 3). 5x5x x 2 + 5x + 6 (x + 3) = 1 Rewrite polynomial as a fraction. = 5x(x + 3) x 2 + 5x + 6 Multiply numerators and denominators. = 5x(x + 3) (x + 3)(x + 2) Factor and divide out common factor. Simplify. = 5x5x x + 2

10. 12.5 Guided Practice Find the product 2w22w2 w 2 – 7w + 12 (w – 4). 4. 2w22w2 w – 3 ANSWER

11. 12.5 Example 4Find the quotient. 7x 2 – 7x x 2 + 2x – 3 x + 1 x 2 – 7x – 8 7x 2 – 7x x 2 + 2x – 3 x + 1 x 2 – 7x – 8 7x 2 – 7x x 2 + 2x – 3x + 1 = x 2 – 7x – 8 Multiply by multiplicative inverse. = (7x 2 – 7x) (x 2 – 7x – 8) (x + 1)(x 2 + 2x – 3) Multiply numerators and denominators. = 7x(x – 1)(x – 8)(x + 1) (x + 3)(x – 1)(x + 1) Factor and divide out common factors. = 7x(x – 8) x + 3 Simplify.

12. 12.5 Guided Practice 5. m 2 – 4 2m 2 + 4m 6m – 3m 2 4m + 44 2(m + 11) 3m23m2 – ANSWER Find the quotient.

13. 12.5 Example 5Find the quotient 2x 2 + 16x + 24 3x23x2 (x + 6). 2x 2 + 16x + 24 3x23x2 (x + 6) = 2x 2 + 16x + 24 3x23x2 (x + 6). = 1 Rewrite polynomial as fraction. 2x 2 + 16x + 24 3x23x2 = 1 (x + 6). Multiply by multiplicative inverse. 2x 2 + 16x + 24 3x23x2 = (x + 6). Multiply numerators and denominators. = 2(x + 2)(x + 6) 3x 2 (x + 6) Factor and divide out common factor. = 2(x + 2) 3x23x2 Simplify.

14. 12.5 Guided Practice 6. n 2 – 6n + 9 12n n – 3 ANSWER 12n n – 3 Find the quotient.

15. 12.5 Lesson Quiz 3x3x x 2 – 3x – 10 (x + 2) ANSWER 3x3x x – 5 2. Find the quotient ( 2x 3 + 11x 2 + 11x – 4 ) ( x + 4) ANSWER 2x 2 + 3x – 1 1. Find the product