If p and q are polynomials with deg(p) < deg(q) and q is a product of distinct linear factors, then the partial fraction decomposition of p(x)/q(x) can be easily computed.
Suppose . Then the partial fraction expansion is
To find c1 “evaluate” the left hand side at x1= b1/a1. Of course, q(x1) = 0, so first factor q(x) and ignore the one zero factor in the denominator. Then (with the zero factor ignored). The other coefficients are found in a similar manner; i.e. set
x = bk/ak, the value that makes the kth factor zero. For example,
The last 73/360 was obtained by evaluating at x = 9 ( ignoring the (x – 9) factor.)
If the denominator q(x) has repeated linear or irreducible quadratic factors then this technique will only work for those coefficients whose denominators are the highest power of each linear term in the partial fraction expansion.
For example, in
only c3 and c5 can be evaluated by the Heaviside method: .
The other coefficients can be found by any of the standard methods.