1. 8.5 Partial Fractions

2. This would be a lot easier if we could re-write it as two separate terms. Multiply by the common denominator. Set like-terms equal to each other. Solve two equations with two unknowns. Example 1

3. Solve two equations with two unknowns. This technique is called Partial Fractions

4. The short-cut for this type of problem is called the Heaviside Method, after English engineer Oliver Heaviside. Multiply by the common denominator. Let x = - 1 Let x = 3

5. 1.If the fraction is improper, use long division to rewrite it as a sum of a polynomial and a proper fraction. 2.If the fraction is proper, factor the denominator completely and write it as a sum of fractions as follows: a) For each linear factors (ax+b) n, the decomposition must have the form: b) ) For each irreducible quadratic factors (ax 2 +bx+c) n, the decomposition must have the form: Method of Partial Fractions

6. Repeated roots: we must use two terms for partial fractions. Example 2

7. If the degree of the numerator is higher than the degree of the denominator, use long division first. (from example one) Example 3

8. irreducible quadratic factor repeated root Example 4

10. Examples