# The Partial Fraction Algorithm

The following theorem is the heart of the partial fraction method. Applied recursively to the denominator, it produces the partial fraction expansion of any rational function.

Theorem 3   Let be a field, and let , be polynomials in such that and are coprime and . Then one may write

with and .

Indeed, as , are coprime, we may apply Theorem 2 and deduce that there exist polynomials , such that

Now let be an arbitrary polynomial (to be fixed later) and define

Then

Now given , Euclid's algorithm (Theorem 1) provides with or such that

and we take this for the definition of our polynomial . It remains to prove that . Suppose, for a contradiction that . Then . Now , so the term dominates and we have that , a contradiction. This proves the theorem.

We now flesh out the algorithm by giving the full partial fraction expansion.

Corollary 1   Let be polynomials such that ; and write , where are mutually pairwise coprime polynomials (that is, no pair of polynomials has a non-trivial common factor), then one can express

for some polynomials such that .

Using our partial fraction algorithm (Theorem 3), a simple induction gives

for some polynomial with or . Consequently, given , with , it is sufficient to express

for some polynomials with .

This is trivial for , and for , we employ Euclid's algorithm to give

with or . Then

As in the proof of Theorem 3, one can show that or . Induction on then gives our result.

Finally, we can employ this result to recover the familiar high school partial fractions result for real polynomials.

Corollary 2   Let be a polynomial over and let be a polynomial over some extension field of (e.g. itself, , etc.). Write

where the are linear polynomials and the are quadratic polynomials, all of which are mutually pairwise coprime. Then there exist constants in the extension field such that

The proof is left to the reader!

Gihan Marasingha 2005-09-19