Partial fraction decomposition expresses a rational function as a polynomial plus a sum
of simpler rational functions whose denominators
are powers of irreducible
polynomials. It is used in integration
by partial fractions. If 
is not a proper
rational function, first apply polynomial
division and decompose the proper rational
function that remains. Over the real numbers,
the irreducible polynomial factors
of 
have polynomial degree 1 or 2 (Bronstein 2005).
For each power 
of a linear factor, introduce
terms
 |
(1)
|
Here 
and 
is a positive integer. For each factor
of the form 
with 
an irreducible
polynomial of polynomial degree 2, introduce
terms
 |
(2)
|
For a proper rational function, write
 |
(3)
|
and solve for the coefficients 
and 
.
If 
is a proper rational function and 
has 
distinct roots 
, 
, ..., 
over the complex numbers,
then
 |
(4)
|
where 
is the derivative of 
(Trott 2006). This follows by writing 
as a Lagrange
interpolating polynomial at the distinct points 
and dividing by 
. For example,
 |
(5)
|
The identities hold wherever the original denominator is nonzero.
Partial fraction decomposition is implemented in the Wolfram Language as Apart.
