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 c_{1 }“evaluate”
the left hand side at x_{1}= b_{1}/a_{1. }Of
course, q(x_{1}) = 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 = b_{k}/a_{k},
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 c_{3 }and c_{5
}can be evaluated by the Heaviside method:
.

The other coefficients can be found by any of the standard methods.