All the results mentioned above are effective, by which I mean that the partial fraction expansions can be actually computed by following the algorithms set out in the theorems and corollaries. The key result is Theorem 2. I give a proof of the corresponding result for integers in the RSA page on my website. The proof in the polynomial case follows in exactly the same manner.
However, in practice, it is often simpler to employ the high school technique of expressing the partial fraction expansion using the unknowns , and , as in Corollary 2, and then applying the `cover-up method' or your favourite trick. The point of my article was to justify the fact that one can produce such an expansion in the first place--a matter which is sadly neglected in the standard curriculum.
Gihan Marasingha 2005-09-19