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.
with and .
Indeed, as , are coprime, we may apply Theorem
2 and deduce that there exist polynomials , such
Now let be an arbitrary polynomial (to be fixed later) and define
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.
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
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.
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