Partial Fraction Expansion
Partial Fraction Expansion or Partial Fraction Decomposition is used to re-write a fraction. One of the possible reasons to do this is to make an expression much easier to integrate.
The Essentials
There are four cases of PFE used for different fractions depending on how factor-able the denominator is:
Case 1: Simple Factors
Case 2: Repeated Factors
Case 3: Simple Irreducible Factors
Case 4: Repeated Irreducible Factors
Note that the fraction that you are re-writing doesn’t have to have just a one for the numerator, it can be any polynomial of order less than the denominator. Also, note that the number of coefficients (A, B, C, etc.) is always equal to the order of the denominator. After getting the fraction into one of these forms multiply the A, B, C, etc by the other terms to get two equal fractions with the same denominator:
Then we set up a system of equations and solve for A, B, C, etc.
Example
Use partial fraction expansion to re-write the fraction:
Practice
Use partial fraction expansion to re-write the fraction: