**General Result**

Let *ƒ*(*x*) be any rational function over the real numbers. In other words, suppose there exist real polynomials *p*(*x*) and *q*(*x*)≠ 0, such that

By removing the leading coefficient of *q*(*x*), we may assume without loss of generality that *q*(*x*) is monic. By the fundamental theorem of algebra, we can write

where *a*_{1},..., *a*_{m}, *b*_{1},..., *b*_{n}, *c*_{1},..., *c*_{n} are real numbers with *b*_{i}2 - 4*c*_{i} < 0, and *j*_{1},..., *j*_{m}, *k*_{1},..., *k*_{n} are positive integers. The terms (*x* - *a*_{i}) are the *linear factors* of *q*(*x*) which correspond to real roots of *q*(*x*), and the terms (*x*_{i}2 + *b*_{i}*x* + *c*_{i}) are the *irreducible quadratic factors* of *q*(*x*) which correspond to pairs of complex conjugate roots of *q*(*x*).

Then the partial fraction decomposition of *ƒ*(*x*) is the following:

Here, *P*(*x*) is a (possibly zero) polynomial, and the *A*_{ir}, *B*_{ir}, and *C*_{ir} are real constants. There are a number of ways the constants can be found.

The most straightforward method is to multiply through by the common denominator *q*(*x*). We then obtain an equation of polynomials whose left-hand side is simply *p*(*x*) and whose right-hand side has coefficients which are linear expressions of the constants *A*_{ir}, *B*_{ir}, and *C*_{ir}. Since two polynomials are equal if and only if their corresponding coefficients are equal, we can equate the coefficients of like terms. In this way, a system of linear equations is obtained which *always* has a unique solution. This solution can be found using any of the standard methods of linear algebra.

Read more about this topic: Partial Fraction, Over The Reals

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