How do you find the partial fraction decomposition when you have repeated quadratic or linear factors?
Imagine the partial fraction decomposition problem:
Here, the denominator would simplify into
However, when the denominator has a repeated factor, something slightly different happens.
However, this will not work. This is remedied by "counting up" to whatever the exponent is on the repeated factor. This would look like
Which can indeed be solved.
An example with a third power:
This is very similar for irreducible quadratic factors, except for that they take the form