partial fractions

Show Summary Details

Quick Reference

Suppose that f(x)/g(x) defines a rational function, so that f(x) and g(x) are polynomials, and suppose that the degree of f(x) is less than the degree of g(x). In general, g(x) can be factorized into a product of some different linear factors, each to some index, and some different irreducible quadratic factors, each to some index. Then the original expression f(x)/g(x) can be written as a sum of terms: corresponding to each (x−α)n in g(x), there are termsand corresponding to each (ax2+bx+c)n in g(x), there are termswhere the real numbers denoted here by capital letters are uniquely determined. The expression f(x)/g(x) is said to have been written in partial fractions. The method, which sounds complicated when stated in general, as above, is more easily understood from examples:The values for the numbers A, B, C,…are found by first multiplying both sides of the equation by the denominator g(x). In the last example, this gives


This has to hold for all values of x, so the coefficients of corresponding powers of x on the two sides can be equated, and this determines the unknowns. In some cases, setting x equal to particular values (in this example, x=1) may determine some of the unknowns more quickly. The method of partial fractions is used in the integration of rational functions.

Subjects: Mathematics.

Reference entries

Users without a subscription are not able to see the full content. Please, subscribe or login to access all content.