Question

There are two ways to find the unknown coefficients of partial fraction decomposition. The second method involves using "nice" values of x that are actually zeros of the denominator. So, the rational function we want to decompose is undefined for them, nevertheless, this method works, why?

Answer 1

We use zeros of the denominator after getting rid of denominators

so, there won't be any denominator on any side

so, it can not be undefined

For example:

We use zeros of denominators after we cancel denominator both sides

so, we can multiply both sides (x-1)(x+1)

now, we can choose zeros of the denominator

It won't be undefined because there is no denominator now

First zero:

Second zero:

now, we can plug it back