---
tag: 030VFEREKG
kind: concept
name: Partial fraction decomposition
msc: [26C15]
sources:
  - url: https://en.wikipedia.org/wiki/Partial_fraction_decomposition
    locator: Basic principles
same_as:
  wikidata: Q431617
  wikipedia: 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|0HH6JX5WZZ]]. Each term $\frac{1}{1 - ax}$ expands as a [[geometric series|0R8JR0ARJ4]].

## Explanations

### L1 human

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