Partial Fractions

EXAMPLE 1: Find


EXAMPLE 2: Find


EXAMPLE 3: Find


EXAMPLE 4: Find


METHOD: Consider a rational function



where and are polynomials. It's possible to express as a sum of simpler fractions provided that the degree of is less than the degree of Such a rational function is called proper. If is improper, then we must take the preliminary step of dividing into (by long division) until a remainder is obtained such that The division statement is



where and are also polynomials.

The next step is to factor the denominator as far as possible. It can be shown that any polynomial can be factored as a product of linear factors (of the form ) and irreducible quadratic factors (of the form where ). For instance,




The third step is to express the proper rational function as a sum of partial fractions of the form



A theorem in algebra guarantees that it is always possible to do this.

There are four possible cases:

Case I: The denominator is a product of distinct linear factors.

This means that we can write



where no factor is repeated (and no factor is a constant multiple of another). For example,



In this case the partial fraction theorem states that there exist constants such that

                                                 (1)




Case II: The denominator is a product of linear factors, some of which are repeated.

This means that we can write



where For example,



In this case instead of the single term in (1), we would use

                                                   (2)




Case III: The denominator contains irreducible quadratic factors, none of which is repeated.

For example,



In this case if has the factor where then, in addition to the partial fractions in (1) and (2), the expression for will have a term of the form

                                                                              (3)




Case IV: The denominator contains a repeated irreducible quadratic factor.

For example,



In this case if has the factor where then, instead of a single partial fraction (3), the sum

                                         (4)

occurs in the partial fraction decomposition of



EXAMPLES:
1. Evaluate


2. Evaluate


3. Evaluate


4. Evaluate


5. Evaluate


6. Evaluate


7. Evaluate


8. Evaluate


9. Evaluate



Appendix A

1. If then



and

2. If then



and

3. If then



and



4. If then



and



5. If then



and




Appendix B

EXAMPLE: Find .