1. Partial Fractions
Lesson 8.5

2. Partial Fraction Decomposition
Consider adding two algebraic fractions
Partial fraction decomposition reverses the process

3. Partial Fraction Decomposition
Motivation for this process
The separate terms are easier to integrate

4. The Process Given Then f(x) can be decomposed with this cascading form
Where polynomial P(x) has degree < n
P(r) ≠ 0
Then f(x) can be decomposed with this cascading form

5. Strategy Given N(x)/D(x)
If degree of N(x) greater than degree of D(x) divide the denominator into the numerator to obtain Degree of N1(x) will be less than that of D(x)
Now proceed with following steps for N1(x)/D(x)

6. Strategy
Factor the denominator into factors of the form where is irreducible
For each factor the partial fraction must include the following sum of m fractions

7. Strategy
Quadratic factors: For each factor of the form , the partial fraction decomposition must include the following sum of n fractions.

8. A Variation Suppose rational function has distinct linear factors
Then we know

9. A Variation
Now multiply through by the denominator to clear them from the equation
Let x = 1 and x = -1
Solve for A and B

10. What If Single irreducible quadratic factor Then cascading form is
But P(x) degree < 2m
Then cascading form is

11. Gotta Try It
Given
Then

12. Gotta Try It Now equate corresponding coefficients on each side
Solve for A, B, C, and D ?

13. Even More Exciting When but Example
P(x) and D(x) are polynomials with no common factors
D(x) ≠ 0
Example

14. Combine the Methods Consider where Express as cascading functions of
P(x), D(x) have no common factors
D(x) ≠ 0
Express as cascading functions of

15. Try It This Time
Given
Now manipulate the expression to determine A, B, and C

16. Partial Fractions for Integration
Use these principles for the following integrals

17. Why Are We Doing This?
Remember, the whole idea is to make the rational function easier to integrate

18. Assignment
Lesson 8.5
Page 559
Exercises 1 – 29 EOO