Definition:Partial Fractions Expansion

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

Let $Q \left({x}\right)$ be expressible as:
 * $Q \left({x}\right) = \displaystyle \prod_{k \mathop = 1}^n q_k \left({x}\right)$

where the $q_k \left({x}\right)$ are themselves polynomial functions of degree at least $1$.

Let $R \left({x}\right)$ be expressible as:
 * $R \left({x}\right) = r \left({x}\right) \displaystyle \sum_{k \mathop = 0}^n \dfrac {p_k \left({x}\right)} {q_k \left({x}\right)}$

where:
 * $r \left({x}\right)$ 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 $p_k \left({x}\right)$ are polynomial functions
 * the degree of $p_k \left({x}\right)$ is strictly less than the degree of $q_k \left({x}\right)$ for all $k$.

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