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

2. Objectives:
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

3. The Partial Fraction Decomposition of a Rational Expression with Distinct Linear Factors in the Denominator

4. Steps in Partial Fraction Decomposition

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. 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. 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. 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. 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. The Partial Fraction Decomposition of Has Repeated Linear Factors

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. 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. 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. 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. 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. 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. The Partial Fraction Decomposition of a Rational Expression with Prime, Nonrepeated Quadratic Factors in the Denominator

18. Example: Partial Fraction Decomposition
Find the partial fraction decomposition of
Step 1 Set up the partial fraction decomposition with the unknown constants.

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. Example: Partial Fraction Decomposition (continued)
Find the partial fraction decomposition of
Step 3 Simplify the right side of the equation.

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. Example: Partial Fraction Decomposition (continued)
Find the partial fraction decomposition of
Step 5 Solve the resulting system for A and B.

23. Example: Partial Fraction Decomposition (continued)
Find the partial fraction decomposition of
Step 5 (cont) Solve the resulting system for A and B.

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. The Partial Fraction Decomposition of a Rational Expression with a Prime, Repeated Quadratic Factor in the Denominator

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. 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. 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. 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. 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. 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.