Partial fraction decomposition

Summary

Partial fraction decomposition writes a rational function as a sum of a polynomial and simple fractions. Let $K$ be a field, and let $P/Q$ be a rational function with $P, Q \in K[X]$ and $Q \neq 0$. Assume that $Q = \prod_{i=1}^{r} q_i^{m_i}$, where the $q_i$ are distinct monic irreducible polynomials. Then there are unique polynomials $B$ and $A_{ij}$, with $\deg A_{ij} < \deg q_i$, such that $$\frac{P}{Q} = B + \sum_{i=1}^{r} \sum_{j=1}^{m_i} \frac{A_{ij}}{q_i^{\,j}}.$$

Remarks

Example: $\frac{1}{1 - x - x^2} = \frac{1}{\sqrt{5}} \left( \frac{\varphi}{1 - \varphi x} - \frac{\psi}{1 - \psi x} \right)$, with $\varphi$ and $\psi$ the roots of $X^2 - X - 1$. In combinatorics it gives the coefficients of a rational generating function. Each term $\frac{1}{1 - ax}$ expands as a geometric series.

Explanations

L1 human

A complicated fraction of polynomials splits into a sum of simple fractions. For example, $\frac{1}{x(x+1)} = \frac{1}{x} - \frac{1}{x+1}$.