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 where and are polynomials, is its expression as where is a polynomial, and, for each , the denominator is a power of an irreducible polynomial (that is not factorable into polynomials of positive degrees), and the numerator 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 be a rational fraction, where and are univariate polynomials in the indeterminate over a field. The existence of the partial fraction can be proved by applying inductively the following reduction steps. Polynomial part There exist two polynomials and such that and where denotes the degree of the polynomial . This results immediately from the Euclidean division of by , which asserts the existence of and such that and This allows supposing in the next steps that Factors of the denominator If and where and are coprime polynomials, then there exist polynomials and such that and This can be proved as follows. Bézout's identity asserts the existence of polynomials and such that (by hypothesis, is a greatest common divisor of and ). Let with be the Euclidean division of by Setting one gets It remains to show that By reducing the last sum of fractions to a common denominator, one gets and thus Powers in the denominator Using the preceding decomposition inductively one gets fractions of the form with where