## 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 (*ax*^{2}+*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

3*x*+2=*A*(*x*−1)(*x*^{2}+*x*+1)+*B*(*x*^{2}+*x*+1)+(*Cx*+*D*)(*x*−1)^{2}.

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.

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