1. Copyright © Cengage Learning. All rights reserved. Rational Expressions and Equations; Ratio and Proportion 6

2. Copyright © Cengage Learning. All rights reserved. Section 6.2 Multiplying and Dividing Rational Expressions

3. 3 Objectives Multiply two rational expressions and write the result in simplest form. Multiply a rational expression by a polynomial and write the result in simplest form. Divide two rational expressions and write the result in simplest form. Divide a rational expression by a polynomial and write the result in simplest form. 1 1 2 2 3 3 4 4

4. 4 Objectives Perform combined operations on three or more rational expressions. 5 5

5. 5 Multiply two rational expressions and write the result in simplest form 1.

6. 6 Multiply two rational expressions and write the result in simplest form Recall, to find the product of and, we proceed as follows. This suggests the rule for multiplying rational expressions. Multiply the numerators and multiply the denominators. Simplify.

7. 7 Multiply two rational expressions and write the result in simplest form Multiplying Rational Expressions If a, b, c, and d are polynomials, then provided no denominators are 0.

8. 8 Example Multiply. Assume that no denominators are 0. a. b. c. d. Solution: We will multiply the numerators, multiply the denominators, and then simplify, if possible. a.

9. 9 Example – Solution b. c. d. cont’d

10. 10 Multiply a rational expression by a polynomial and write the result in simplest form 2.

11. 11 Multiply a rational expression by a polynomial and write the result in simplest form Since any number divided by 1 remains unchanged, we can write any polynomial as a rational expression by writing it with a denominator of 1.

12. 12 Example Multiply:. Assume that the denominator is not 0. Solution: We will write x + 7 as, multiply the numerators, multiply the denominators, and then simplify. Write x + 7 as a fraction with a denominator of 1.

13. 13 Example – Solution Divide out all common factors. Multiply the fractions and factor where possible. cont’d

14. 14 Divide two rational expressions and write the result in simplest form 3.

15. 15 Divide two rational expressions and write the result in simplest form Recall that division by a nonzero number is equivalent to multiplying by the reciprocal of that number. Thus, to divide two fractions, we can invert the divisor (the fraction following the  sign) and multiply. For example, to divide by, we proceed as follows: This suggests the rule for dividing rational expressions. Invert and write as multiplication. Multiply the numerators and multiply the denominators.

16. 16 Divide two rational expressions and write the result in simplest form Dividing Rational Expressions If a, b, c, and d are polynomials, then provided no denominators are equal to 0.

17. 17 Example Divide, assuming that no denominators are 0. a. b.

18. 18 Example a) – Solution We will change each division to a multiplication and then multiply the resulting rational expressions. Divide out common factors. Invert the divisor and multiply.

19. 19 Example b) – Solution Invert the divisor and multiply. Divide out common factors. Multiply the remaining factors. cont’d

20. 20 Divide a rational expression by a polynomial and write the result in simplest form 4.

21. 21 Divide a rational expression by a polynomial and write the result in simplest form To divide a rational expression by a polynomial, we write the polynomial as a rational expression by inserting a denominator of 1 and then divide the expressions.

22. 22 Example Divide:. Assume that no denominators are 0. Solution: We will write 4 – x 2 as, invert the divisor, and then multiply the resulting rational expressions. Write 4 – x 2 as a fraction with a denominator of 1.

23. 23 Example 8 – Solution Invert the divisor and multiply. Factor where possible. Divide out common factors: = –1. Multiply. OR cont’d

24. 24 Perform combined operations on three or more rational expressions 5.

25. 25 Perform combined operations on three or more rational expressions Unless parentheses indicate otherwise, we will perform multiplications and divisions in order from left to right.

26. 26 Example Simplify:. Assume that no denominators are 0. Solution: Since there are no parentheses to indicate otherwise, we perform the division first.

27. 27 Example – Solution Invert the divisor and multiply. Factor where possible and multiply. Divide out all common factors. cont’d