In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator. The importance of the partial fraction decomposition lies in the fact that it provides algorithms for various computations with rational functions, including the explicit computation of antiderivatives, Taylor series expansions, inverse Z-transforms, and inverse Laplace transforms. The concept was discovered independently in 1702 by both Johann Bernoulli and Gottfried Leibniz. In symbols, the partial fraction decomposition of a rational fraction of the form f ( x ) g ( x ) , {\textstyle {\frac {f(x)}{g(x)}},} where f and g are polynomials, is the expression of the rational fraction as
f ( x ) g ( x ) = p ( x ) + ∑ j f j ( x ) g j ( x ) {\displaystyle {\frac {f(x)}{g(x)}}=p(x)+\sum _{j}{\frac {f_{j}(x)}{g_{j}(x)}}}
where p(x) is a polynomial, and, for each j, the denominator gj (x) is a power of an irreducible polynomial (i.e. not factorizable into polynomials of positive degrees), and the numerator fj (x) is a polynomial of a smaller degree than the degree of this irreducible polynomial. When explicit computation is involved, a coarser decomposition is often preferred, which consists of replacing "irreducible polynomial" by "square-free polynomial" in the description of the outcome. This allows replacing polynomial factorization by the much easier-to-compute square-free factorization. This is sufficient for most applications, and avoids introducing irrational coefficients when the coefficients of the input polynomials are integers or rational numbers.
Basic principles Let
R ( x ) = F G {\displaystyle R(x)={\frac {F}{G}}} be a rational fraction, where F and G are univariate polynomials in the indeterminate x over a field. The existence of the partial fraction decomposition can be proved by applying inductively the following reduction steps.
Polynomial part There exist two polynomials E and F1 such that
F G = E + F 1 G , {\displaystyle {\frac {F}{G}}=E+{\frac {F_{1}}{G}},}
and
deg F 1 < deg G , {\displaystyle \deg F_{1}<\deg G,}
where deg P {\displaystyle \deg P} denotes the degree of the polynomial P. This results immediately from the Euclidean division of F by G, which asserts the existence of E and F1 such that F = E G + F 1 {\displaystyle F=EG+F_{1}} and deg F 1 < deg G . {\displaystyle \deg F_{1}<\deg G.}
This allows supposing in the next steps that deg F < deg G . {\displaystyle \deg F<\deg G.}
Factors of the denominator If deg F < deg G , {\displaystyle \deg F<\deg G,} and
G = G 1 G 2 , {\displaystyle G=G_{1}G_{2},}
where G1 and G2 are coprime polynomials, then there exist polynomials F 1 {\displaystyle F_{1}} and F 2 {\displaystyle F_{2}} such that
F G = F 1 G 1 + F 2 G 2 , {\displaystyle {\frac {F}{G}}={\frac {F_{1}}{G_{1}}}+{\frac {F_{2}}{G_{2}}},}
and
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