Partial Fraction Decomposition

Topics: Calculus - Integrals

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Theorem 3 §

Let $\frac{p(x)}{q(x)}$ be a rational function with $\mathrm{deg}(p(x))<\mathrm{deg}(q(x))$, $(p(x),q(x))=1$, and such that $q(x)=(a_1{x}^2+b_{1}x+c_1)+\cdots +(a_n{x}^2+b_{n}x+c_n)$, where each $a_i{x}^2+b_{i}x+c_i$ is an irreducible quadratic polynomial.

Then, there exist $A_1,B_1,\dots ,A_n,B_n\in \mathbb{R}$ such that:

$$\frac{p(x)}{q(x)}=\frac{A_{1}x+B_1}{a_1{x}^2+b_{1}x+c_1}+\frac{A_{2}x+B_2}{a_2{x}^2+b_{2}x+c_2}+\cdots +\frac{A_{n}x+B_n}{a_n{x}^2+b_{n}x+c_n}$$