1. Chapter 7 Systems of Equations and Inequalities Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 7.3 Partial Fractions

2. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 Decompose where Q has only distinct linear factors. Decompose where Q has repeated linear factors. Decompose where Q has a nonrepeated prime quadratic factor. Decompose where Q has a prime, repeated quadratic factor. Objectives:

3. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3 The Partial Fraction Decomposition of a Rational Expression with Distinct Linear Factors in the Denominator

4. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4 Steps in Partial Fraction Decomposition

5. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5 Example: Partial Fraction Decomposition with Distinct Linear Factors Find the partial fraction decomposition of Step 1 Set up the partial fraction decomposition with the unknown constants.

6. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6 Example: Partial Fraction Decomposition with Distinct Linear Factors (continued) Find the partial fraction decomposition of Step 2 Multiply both sides of the resulting equation by the least common denominator.

7. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7 Example: Partial Fraction Decomposition with Distinct Linear Factors (continued) Find the partial fraction decomposition of Step 3 Simplify the right side of the equation. Step 4 Write both sides in descending powers, equate coefficients of like powers of x, and equate constant terms.

8. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8 Example: Partial Fraction Decomposition with Distinct Linear Factors (continued) Find the partial fraction decomposition of Step 5 Solve the resulting system for A and B.

9. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9 Example: Partial Fraction Decomposition with Distinct Linear Factors (continued) Find the partial fraction decomposition of Step 6 Substitute values of A and B and write the partial fraction decomposition. A = 2, B = 3

10. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10 The Partial Fraction Decomposition of Has Repeated Linear Factors

11. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11 Example: Partial Fraction Decomposition with Repeated Linear Factors Find the partial fraction decomposition of Step 1 Set up the partial fraction decomposition with the unknown constants.

12. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12 Example: Partial Fraction Decomposition with Repeated Linear Factors (continued) Find the partial fraction decomposition of Step 2 Multiply both sides of the resulting equation by the least common denominator.

13. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13 Example: Partial Fraction Decomposition with Repeated Linear Factors (continued) Find the partial fraction decomposition of Step 3 Simplify the right side of the equation.

14. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 14 Example: Partial Fraction Decomposition with Repeated Linear Factors (continued) Find the partial fraction decomposition of Step 4 Write both sides in descending powers, equate coefficients of like powers of x, and equate constant terms.

15. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 15 Example: Partial Fraction Decomposition with Repeated Linear Factors (continued) Find the partial fraction decomposition of Step 5 Solve the resulting system for A and B.

16. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 16 Example: Partial Fraction Decomposition with Repeated Linear Factors (continued) Find the partial fraction decomposition of Step 6 Substitute values of A and B and write the partial fraction decomposition.

17. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 17 The Partial Fraction Decomposition of a Rational Expression with Prime, Nonrepeated Quadratic Factors in the Denominator

18. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 18 Example: Partial Fraction Decomposition Find the partial fraction decomposition of Step 1 Set up the partial fraction decomposition with the unknown constants.

19. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 19 Example: Partial Fraction Decomposition (continued) Find the partial fraction decomposition of Step 2 Multiply both sides of the resulting equation by the least common denominator.

20. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 20 Example: Partial Fraction Decomposition (continued) Find the partial fraction decomposition of Step 3 Simplify the right side of the equation.

21. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 21 Example: Partial Fraction Decomposition (continued) Find the partial fraction decomposition of Step 4 Write both sides in descending powers, equate coefficients of like powers of x, and equate constant terms.

22. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 22 Example: Partial Fraction Decomposition (continued) Find the partial fraction decomposition of Step 5 Solve the resulting system for A and B.

23. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 23 Example: Partial Fraction Decomposition (continued) Find the partial fraction decomposition of Step 5 (cont) Solve the resulting system for A and B.

24. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 24 Example: Partial Fraction Decomposition (continued) Find the partial fraction decomposition of Step 6 Substitute values of A and B and write the partial fraction decomposition.

25. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 25 The Partial Fraction Decomposition of a Rational Expression with a Prime, Repeated Quadratic Factor in the Denominator

26. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 26 Example: Partial Fraction Decomposition with a Repeated Quadratic Factor Find the partial fraction decomposition of Step 1 Set up the partial fraction decomposition with the unknown constants.

27. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 27 Example: Partial Fraction Decomposition with a Repeated Quadratic Factor (continued) Find the partial fraction decomposition of Step 2 Multiply both sides of the resulting equation by the least common denominator.

28. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 28 Example: Partial Fraction Decomposition with a Repeated Quadratic Factor (continued) Find the partial fraction decomposition of Step 3 Simplify the right side of the equation.

29. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 29 Example: Partial Fraction Decomposition with a Repeated Quadratic Factor (continued) Find the partial fraction decomposition of Step 4 Write both sides in descending powers, equate coefficients of like powers of x, and equate constant terms.

30. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 30 Example: Partial Fraction Decomposition with a Repeated Quadratic Factor (continued) Find the partial fraction decomposition of Step 5 Solve the resulting system for A and B.

31. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 31 Example: Partial Fraction Decomposition with a Repeated Quadratic Factor (continued) Find the partial fraction decomposition of Step 6 Substitute values of A and B and write the partial fraction decomposition.