## 175- Partial fractions decomposition

November 18, 2008

[This post replaces the previous version from October 12, 2008. The new argument is significantly simpler than the one originally posted.]

I want to present here a Calculus II-level proof that the method of partial fractions decomposition works. I will actually show a more general result, which will greatly simplify the presentation and get rid of most problems of the somewhat awkward formulation below.

We need some notation:

• $\prod_{i=1}^n a_i$ means $a_1\times a_2\times\dots\times a_n$.
• $p(x)$, $q(x)$, etc, denote polynomials with real coefficients.

The proof that I show below is algorithmic in nature, meaning that it provides us with a method to find the relevant constants for any given specific polynomials $p$ and $q$. The constants we obtain are real numbers.

That the method of partial fractions decomposition works means that we can always find the relevant constants the method requires.