Partial fraction decomposition

Any rational function of the form $\frac{P(x)}{Q(x)}$ maybe written as a sum of simpler rational functions. This allows for much easier integration, as well as causing many sums to telescope. It has many other uses, as well.

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 $\frac{R(x)}{Q(x)}$ .

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Partial fraction decomposition