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:

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

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


Let $\map R x$ be expressible as:

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

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


Also known as

A partial fractions expansion is also known as a decomposition (of a fraction).


Examples

Example: $\frac 5 6$

$\dfrac 5 6$ has the partial fractions expansion given by:

$\dfrac 5 6 = \dfrac 1 2 + \dfrac 1 3$

Hence $\dfrac 1 2$ and $\dfrac 1 3$ are partial fractions of $\dfrac 5 6$.


Arbitrary Example $1$: $\dfrac 1 {x \paren {x + 1}^2}$

$\dfrac 1 {x \paren {x + 1}^2} = \dfrac 1 x - \dfrac 1 {\paren {x + 1} } - \dfrac 1 {\paren {x + 1}^2}$


Arbitrary Example $2$: $\dfrac {x + 3} {x^2 + 3 x + 2}$

$\dfrac {x + 3} {x^2 + 3 x + 2} = \dfrac 2 {x + 1} -\dfrac 1 {x + 2}$


Also see

  • Results about partial fractions expansions can be found here.


Sources

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