1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Digital Lesson on Operations on Rational Expressions

2. Slide 13- 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley To multiply rational expressions: 1. Write each numerator and denominator in factored form. 2.Divide out any numerator factor with any matching denominator factor. 3. Multiply numerator by numerator and denominator by denominator. 4. Simplify as needed. Example: Multiply Solution:

3. Slide 13- 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Multiply. Solution Write numerators and denominators in factored form. Multiply the remaining numerator factors and denominator factors.

4. Slide 13- 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Dividing Rational Expressions To divide rational expressions: 1. Write an equivalent multiplication statement with the reciprocal of the divisor. 2.Write each numerator and denominator in factored form. (Steps 1 and 2 are interchangeable.) 3. Divide out any numerator factor with any matching denominator factor. 4. Multiply numerator by numerator and denominator by denominator. 5.Simplify as needed.

5. Slide 13- 5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Divide. Solution Write an equivalent multiplication statement. Divide out common factors, and multiply remaining factors.

6. Slide 13- 6 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Adding or Subtracting Rational Expressions (Same Denominator) To add or subtract rational expressions that have the same denominator: 1. Add or subtract the numerators and keep the same denominator. 2. Simplify to lowest terms (remember to write the numerators and denominators in factored form in order to simplify).

7. Slide 13- 7 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Add. Solution Since the rational expressions have the same denominator, we add numerators and keep the same denominator. Factor. Divide out the common factor, 2.

8. Slide 13- 8 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Subtract. Solution Divide out the common factor, x – 4. Note: The numerator can be factored, so we may be able to simplify.

9. Slide 13- 9 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Add. Solution Combine like terms in the numerator. Factor the numerator and the denominator. Divide out the common factor, b.

10. Slide 13- 10 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Add. Solution Combine like terms in the numerator. Factor the numerator and the denominator.

11. Slide 13- 11 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley continued Divide out the common factors, 2 and x + 2.

12. Slide 13- 12 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Subtract. Solution Note: To write an equivalent addition, change the operation symbol from a minus sign to a plus sign and change all the signs in the subtrahend (second) polynomial.

13. Slide 13- 13 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Finding the LCD To find the LCD of two or more rational expressions: 1. Factor each denominator. 2. For each unique factor, compare the number of times it appears in each factorization. Write a product that includes each unique factor the greatest number of times it appears in the denominator factorizations.

14. Slide 13- 14 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Find the LCD. Solution We first factor the denominators 12y 2 and 8y 3 by writing their prime factorizations. The unique factors are 2, 3, and y. To generate the LCD, include 2, 3, and y the greatest number of times each appears in any of the factorizations.

15. Slide 13- 15 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley continued Note: We can compare exponents in the prime factorizations to create the LCD. If two factorizations have the same prime factors, we write that prime factor in the LCD with the greater of the two exponents. The greatest number of times that 2 appears is three times (in 2 3 y 3 ). The greatest number of times that 3 appears is once (in 2 2 3 y 2 ). The greatest number of times that y appears is three times (in 2 3 y 3 ).

16. Slide 13- 16 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Find the LCD. Solution Factor the denominators x 2 – 25 and 2x – 10. The unique factors are 2, (x + 5), and (x – 5). The greatest number of times that 2 appears is once. The greatest number of times that (x + 5) appears is once. The greatest number of times that (x – 5) appears is once.

17. Slide 13- 17 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Adding or Subtracting Rational Expressions with Different Denominators To add or subtract rational expressions with different denominators: 1. Find the LCD. 2. Write each rational expression as an equivalent expression with the LCD. 3. Add or subtract the numerators and keep the LCD. 4. Simplify.

18. Slide 13- 18 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Add. Solution The LCD is 24x 2. Write equivalent rational expressions with the LCD, 24x 2. Add numerators. Note: Remember that to add polynomials, we combine like terms.

19. Slide 13- 19 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Add. Solution Since x – 6 and 6 – x are additive inverses, we obtain the LCD by multiplying the numerator and denominator of one of the rational expressions by –1. We chose the second rational expression.

20. Slide 13- 20 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Complex rational expression: A rational expression that contains rational expressions in the numerator or denominator. Examples:

21. Slide 13- 21 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Simplifying Complex Rational Expressions To simplify a complex rational expression, use one of the following methods: Method 1 1. Simplify the numerator and denominator if needed. 2. Rewrite as a horizontal division problem. Method 2 1. Multiply the numerator and denominator of the complex rational expression by their LCD. 2. Simplify.

22. Slide 13- 22 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example—Method 1 Simplify. Solution Write the numerator fractions as equivalent fractions with their LCD, 12, and write the denominator fractions with their LCD, 24.

23. Slide 13- 23 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley continued 1 2

24. Slide 13- 24 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example—Method 2 Simplify. Solution Multiply the numerator and denominator by 24. 1 46 1 3 1 4 1

25. Slide 13- 25 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Simplify. Solution Write the numerator and denominator as equivalent fractions. (Method 1)