Partial Fraction Decomposition

Partial fraction decomposition is a method of breaking down fractions of polynomials into more lightweight easier to evaluate fractions.

Given an arbitrary polynomial fraction of the structure

$$f(x)=\frac{P(x)}{Q(x)}$$

The requirement for partial decomposition is that the degree of $Q(x)$ is larger than the degree of $P(x)$.

Workflow

First we factor the denominator $Q(x)$ as far as possible to reach

$$\frac{A}{(a\_1x+b\_1)}+\frac{B}{(a\_2x+b\_2)}+\dots$$

We then cross multiply our nominator values $A,B,\dots$ such that we receive

$$A(a\_2x+b\_2)\text{and}B(a\_1x+b\_1)$$

Then set our nominators multiplied by the factors of the factored denominator equal to the nominator of the initial expression

$$P(x)=A(a\_1x+b)+B(a\_2x+b\_2)+\dots$$

And solve the system of equations for A and B and $\dots$ by plugging in the roots of our initially factored denominator $(a\_1x+b\_1)(a\_2x+b\_2)\dots$ into $P(x)$. We then plug the values of A, B and $\dots$ back into the expression

$$\frac{A}{(a\_1x+b\_1)}+\frac{B}{(a\_2x+b\_2)}+\dots$$

We have now factored $f(x)$ by using partial fractions.