Partial fraction decompostion is a theorem in algebra which says that a rational function can be decomposed into a polynomial plus a sum of proper fractions, each of which is either a constant over a power of a linear polynomial or a linear polynomial over a power of an irreducible quadratic polynomial. Algebra is a branch of mathematics concerning the study of structure, relation and quantity. ... In mathematics, a rational function in algebra is a function defined as a ratio of polynomials. ...

Statement of theorem

Let f and g be nonzero polynomials. Write g as a product of powers of distinct irreducible polynomials:

There are (unique) polynomials b and a_ij with such that

If , then b = 0.

Outline of proof

Lemma 1

Let f,g and h be nonzero polynomials with f and g coprime. There are polynomials a and b such that Coprime - Wikipedia /**/ @import /skins-1. ...

Lemma 2

Let f and g be nonzero polynomials and let n be a positive integer. There exist polynomials b and a_i with such that

Generalization to Euclidean domains

More generally, this is true in any Euclidean domain. In abstract algebra, a Euclidean domain (also called a Euclidean ring) is a type of ring in which the Euclidean algorithm can be used. ...