# 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*

Gihan Marasingha 2005-09-19