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  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) 1.If degree of N(x) greater than degree of D(x) divide the denominator into the numerator to obtain Degree of N 1 (x) will be less than that of D(x)  Now proceed with following steps for N 1 (x)/D(x)

6. Strategy 2.Factor the denominator into factors of the form where is irreducible 3.For each factor the partial fraction must include the following sum of m fractions

7. Strategy 4.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  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  P(x) and D(x) are polynomials with no common factors  D(x) ≠ 0 Example

14. Combine the Methods Consider where  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 – 45 EOO