# Continued Fraction summation representation

I have a rational fraction of the form:

$$s=\frac{p_0+p_1x+p_2x^2+\cdots+p_Mx^M}{1+q_1x+q_2x^2+\cdots+q_Mx^M}$$

The paper I am reading converts this to the form:

$$s = b_0+\sum^M_{m=1}\frac{b_m}{\lambda_m+x}$$

But it doesn't show any steps. Does anyone know the steps for how they got the second equation?

Any help is greatly appreciated.

Cheers

• en.wikipedia.org/wiki/Partial_fraction_decomposition – Patissot Mar 2 '15 at 22:42
• Is there some way to do this without factoring the denominator? I am dealing with M = 50+ and the Wikipedia article you linked to seems to suggest that factoring the denominator is always necessary. – Darcy Mar 2 '15 at 23:00