# Definition:Partial Fractions Expansion

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## Definition

Let $\map R x = \dfrac {\map P x} {\map Q x}$ be a rational function, where $\map P x$ and $\map Q x$ are expressible as polynomial functions.

Let $\map Q x$ be expressible as:

- $\map Q x = \displaystyle \prod_{k \mathop = 1}^n \map {q_k} x$

where the $\map {q_k} x$ are themselves polynomial functions of degree at least $1$.

Let $\map R x$ be expressible as:

- $\map R x = \map r x \displaystyle \sum_{k \mathop = 0}^n \dfrac {\map {p_k} x} {\map {q_k} x}$

where:

- $\map r x$ is a polynomial function which may or may not be the null polynomial, or be of degree $0$ (that is, a constant)
- each of the $\map {p_k} x$ are polynomial functions
- the degree of $\map {p_k} x$ is strictly less than the degree of $\map {q_k} x$ for all $k$.

Then $\map r x \displaystyle \sum_{k \mathop = 0}^n \dfrac {\map {p_k} x} {\map {q_k} x}$ is a **partial fractions expansion** of $\map R x$.

## Sources

- 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.20$: The Bernoulli Numbers and some Wonderful Discoveries of Euler