1. College Algebra Chapter 5 Systems of Equations and Inequalities
Section 5.3
Partial Fraction Decomposition

2. Concepts
1. Set up a Partial Fraction Decomposition
2. Decompose , where is a Product of Linear Factors
3. Decompose , where has Irreducible Quadratic Factors

3. Set up a Partial Fraction Decomposition
Previously, we have added or subtracted rational expressions.
For example,

4. Set up a Partial Fraction Decomposition
Now we will learn the technique of partial fraction decomposition to write a rational expression as the sum of simpler fractions.
In this case, we start with the rational expression
and decompose it into two simpler
fractions.

5. Example:
Find constants A and B such that
Multiply both sides by the common denominator

6. Example continued:
Simplify and combine like terms.
Two polynomials are equal if and only if the coefficients on like terms are equal.

7. Example continued:
Solve the system of linear equations.

8. Set up a Partial Fraction Decomposition
How to begin a partial fraction decomposition?
Make sure the degree of the numerator is less than the degree of the denominator.
Factor the denominator. The denominator will factor into linear factors (ax + b) and quadratic factors (ax2 + bx + c) that are irreducible over the integers.

9. Set up a Partial Fraction Decomposition
There will be one (or more) of four types:

10. Example 1:
Set up the form for the partial fraction decomposition for the given rational expressions.
(distinct linear factors)

11. Example 2:
Set up the form for the partial fraction decomposition for the given rational expressions.
(repeated linear factors)

12. Example 3:
Set up the form for the partial fraction decomposition for the given rational expressions.
(repeated linear factors)

14. Example 4:
Set up the form for the partial fraction decomposition for the given rational expressions.
(linear/quadratic factors)

15. Example 5:
Set up the form for the partial fraction decomposition for the given rational expressions.
(repeated linear/quadratic factors)

17. Concepts
1. Set up a Partial Fraction Decomposition
2. Decompose , where is a Product of Linear Factors
3. Decompose , where has Irreducible Quadratic Factors

18. Example 6:
Find the partial fraction decomposition.

19. Example 6 continued:

20. Example 6 continued:

21. Example 7:
Find the partial fraction decomposition.

22. Example 7 continued:

25. Concepts
1. Set up a Partial Fraction Decomposition
2. Decompose , where is a Product of Linear Factors
3. Decompose , where has Irreducible Quadratic Factors

26. Example 8:
Find the partial fraction decomposition.

27. Example 8 continued:

28. Example 8 continued:

29. Example 9:
Find the partial fraction decomposition.

30. Example 9 continued: