Partial fraction decomposition

Any rational function of the form $\frac{P(x)}{Q(x)}$ maybe written as a sum of simpler rational functions.

To find the decomposition of a rational function, first perform the long division operation on it. This transforms the function into one of the form $\frac{P(x)}{Q(x)}=S(x) + \frac{R(x)}{Q(x)}$ , where $R(x)$ is the remainder term and $\deg R(x) \leq \deg Q(x)$ .

Next, for every factor $(a_nx^n+a_{n-1}x^{n-1}+\ldots +a_0)^m$ in the factorization of $Q(x)$ , introduce the terms

$\frac{A_1x^{n-1}+B_1x^{n-2}+\ldots+Z_1}{a_nx^n+a_{n-1}x^{n-1}+\ldots+a_0}+\frac{A_2x^{n-1}+B_2x^{n-2}+\ldots+Z_2}{(a_nx^n+a_{n-1}x^{n-1}+\ldots+a_0)^2}+\ldots+\frac{A_mx^{n-1}+B_mx^{n-2}+\ldots+Z_m}{(a_nx^n+a_{n-1}x^{n-1}+\ldots+a_0)^m}$

(Note that the variable $Z_i$ has no relation to being the 26th letter in the alphabet.)

Next, take the sum of every term introduced above and equate it to $\frac{R(x)}{Q(x)}$ , and solve for the variables $A_i, B_i, \ldots$ . Once you solve for all the variables, then you will have the partial fraction decomposition of $\frac{R(x)}{Q(x)}$ .

Partial fraction decomposition has several common uses. It allows for much easier integration of rational functions, allowing one to integrate a complicated rational function term by term. Additionally, it can be used in summations, causing sums to often telescope or have a much easier form which can be expressed with a closed form. It has several other minor uses, as well.

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Partial fraction decomposition