<s>[INST] Partial fraction decomposition with a 3rd degree numerator I have the following function to decompose using PFD:
$H(x) = \frac{x^3 + 4x^2 - 11x - 48}{x^3 + 6x^2 + 3x - 10}$ 
The poles are $1$, $-2$ and $-5$ so I tried to do it this way: finding $a$, $b$ and $c$ such as:
$H(x) = \frac{a}{x-1} + \frac{b}{x+2} + \frac{c}{x+5}$ 
But it didn't give me a correct result. Doing it that way gives me $(a,b,c) = (-3, 2, -1) $
Is there a specific form of numerator to introduce here? Thanks.
 [/INST] Since the degree of the numerator is $\geq$ to the degree of the denominator you should first divide them. Hence
$$\frac{x^3 + 4x^2 - 11x - 48}{x^3 + 6x^2 + 3x - 10}=1-\frac{2x^2 + 14x + 38}{x^3 + 6x^2 + 3x - 10}=1+\frac{a}{x-1} + \frac{b}{x+2} + \frac{c}{x+5}$$
for some real constants $a$, $b$, $c$ (the same that you already found by using the residue method!).
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