1. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1 Section 6.3 Complex Rational Expressions Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1

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3. 3 Simplifying Complex Fractions Complex rational expressions, also called complex fractions, have numerators or denominators containing one or more fractions. Woe is me, for I am complex.

4. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 4 Complex Rational Expressions Simplifying a Complex Rational Expression by Multiplying by 1 in the Form T 1) Find the LCD of all rational expressions within the complex rational expression. 2) Multiply both the numerator and the denominator of the complex rational expression by this LCD. 3) Use the distributive property and multiply each term in the numerator and denominator by this LCD. Simplify each term. No fractional expressions should remain within the numerator and denominator of the main fraction. 4) If possible, factor and simplify.

5. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 5 Simplifying Complex FractionsEXAMPLE Simplify: SOLUTION The denominators in the complex rational expression are 5 and x. The LCD is 5x. Multiply both the numerator and the denominator of the complex rational expression by 5x. Multiply the numerator and denominator by 5x.

6. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 6 Simplifying Complex Fractions Use the distributive property. CONTINUED Divide out common factors. Simplify. Factor and simplify.

7. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 7 Simplifying Complex Fractions Simplify. CONTINUED

8. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 8 Simplifying Complex FractionsEXAMPLE Simplify: SOLUTION The denominators in the complex rational expression are x + 6 and x. The LCD is (x + 6)x. Multiply both the numerator and the denominator of the complex rational expression by (x + 6)x. Multiply the numerator and denominator by (x + 6)x.

9. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 9 Simplifying Complex Fractions Use the distributive property. CONTINUED Divide out common factors. Simplify.

10. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 10 Simplifying Complex FractionsCONTINUED Subtract. Factor and simplify. Simplify.

11. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 11 Objective #1: Example

12. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 12 Objective #1: Example

13. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 13 Objective #1: Example

14. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 14 Objective #1: Example

15. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 15

16. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 16 Simplifying Complex Fractions Simplifying a Complex Rational Expression by Dividing 1) If necessary, add or subtract to get a single rational expression in the numerator. 2) If necessary, add or subtract to get a single rational expression in the denominator. 3) Perform the division indicated by the main fraction bar: Invert the denominator of the complex rational expression and multiply. 4) If possible, simplify.

17. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 17 Simplifying Complex FractionsEXAMPLE Simplify: SOLUTION 1) Subtract to get a single rational expression in the numerator.

18. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 18 Simplifying Complex Fractions 2) Add to get a single rational expression in the denominator. CONTINUED

19. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 19 Simplifying Complex Fractions 3) & 4) Perform the division indicated by the main fraction bar: Invert and multiply. If possible, simplify. CONTINUED

20. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 20 Objective #2: Example

21. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 21 Objective #2: Example

22. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 22 Objective #2: ExampleCONTINUED

23. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 23 Objective #2: ExampleCONTINUED

24. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 24 Objective #2: Example

25. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 25 Objective #2: Example

26. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 26 Objective #2: ExampleCONTINUED

27. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 27 Objective #2: ExampleCONTINUED