1. Chapter 6 Section 2 Multiplication and Division of Rational Expressions 1

2. Multiplying Rational Expressions Multiplying Two Fractions Example: or we can divide out common factors first then multiply. 2

3.  Factor the numerator and the denominator completely.  Divide out common factors.  Multiply numerators together and multiply denominators together.  Note: Factor -1 when numerator and denominator only differ by their signs  Rule: Multiplying Rational Expressions 3

4. Example: Multiplying Rational Expressions 4

5. Example: Multiplying Rational Expressions Factor out common factor of x 2 5

6. Example: Multiplying Rational Expressions We leave the numerator as a polynomial (in unfactored form) and the denominator in factored form. Write as two factors Difference in Two Squares: a 2 – b 2 = (a + b)(a – b) a = x and b = 3 x 2 – 3 2 = (x + 3)(x – 3) 6

7. Example: Multiplying Rational Expressions When only the signs differ in a numerator and denominator factor out -1 (in either the numerator or denominator) then divide out the common factor. 7

8. Example: Multiplying Rational Expressions Factor out -1 then GCF of 2 8

9. Example: Multiplying Rational Expressions (a)(c) = (3)(-10)=-30 Factors of -30 that sum to 13 (15)(-2) = -30 and (15)+(-2) = 13 (a)(c) = (2)(-1)=-2 Factors of -2 that sum to 1 (2)(-1) = -2 and (2)+(-1) = 1 (a)(c) = (3)(-2)=-6 Factors of -6 that sum to 1 (3)(-2) = -6 and (3)+(-2) = 1 (a)(c) = (2)(-5)=-10 Factors of -10 that sum to 9 (-1)(10) = -10 and (-1)+(10) = 9 9

10. Example: Multiplying Rational Expressions 2a 2 – 9a + 4 (a)(c) = (2)(4) = 8 Factors of 8 that sum to -9 (-8)(-1) = 8 and (-8)+(-1) = -9 Replace -9a with factors (-8) and (-1) then factor by grouping. 10a 2 – 5a Factor out the common factor of 5a -4a Factor out a -1 10

11. Example: Multiplying Rational Expressions x 2 – y 2 Difference in Two Squares: a 2 – b 2 = (a + b)(a – b) a = x and b = y x 2 – y 2 = (x + y)(x – y) 3x 2 + 4xy + y 2 (a)(c) = (3)(1) = 3 Factors of 3 that sum to 4 (3)(1) = 3 and (3)+(1) = 4 Replace 4xy with the factors 3 and 1 Factor by grouping 11

12. Divide Rational Expressions Rule: Divide Two Fractions Example: Invert the devisor (the second fraction) and multiply 12

13. Example: To Divide a Rational Expression we multiply the first fraction by the reciprocal of the second fraction. Divide Rational Expressions 13

14. Example: To Divide a Rational Expression we multiply the first fraction by the reciprocal of the second fraction. Divide Rational Expressions Difference in Two Squares: a 2 – b 2 = (a + b)(a – b) a = x and b = 4 x 2 – 4 2 = (x + 4)(x – 4) 14

15. Example: Divide Rational Expressions When only the signs differ in a numerator and denominator factor out a -1 (in either the numerator or denominator) then divide out the common factor. 15

16. Example: Divide Rational Expressions X 2 + 3x – 18 Factors of -18 that sum to 3 (6)(-3) = -18 and (6)+(-3) = 3 16

17. Example: Divide Rational Expressions (a)(c) = (2)(-12) = -24 Factors of -24 that sum to 5 (8)(-3) = -24 and (8)+(-3) = 5 Factor out any common factors (a)(c) = (2)(-3) = -6 Factors of -6 that sum to -1 (2)(-3) = -6 and (2)+(-3) = -1 17

18. Special Factoring Difference of Two Squares Sum of Two Cubes Difference of Two Cubes 18

19. Remember  Factor completely before you simplify.  Change a division problem to a multiplication problem before factoring and canceling.  Factor out a -1 when the terms only differ by the signs.  Special factoring makes the problem easier. 19

20. HOMEWORK 6.2 Page 366-367: # 19, 23, 27, 33, 39, 45, 63, 87 20