<s>[INST] Why partial fraction decomposition of $\frac{1}{s^2(s+2)}$ is $\frac{A}{s}+\frac{B}{s^2}+\frac{C}{(s+2)}$? Can someone please explain why: $$\frac{1}{s^2(s+2)}=\frac{A}{s}+\frac{B}{s^2}+\frac{C}{(s+2)}$$
And not:$$\frac{1}{s^2(s+2)}=\frac{A}{s^2}+\frac{B}{(s+2)}$$
I'm a bit confused where the extra s term comes from in the first equation.
 [/INST] You can do it step-by-step:
$$\begin{align}\frac{1}{s^2(s+2)}&=\frac1s\cdot \frac{1}{s\cdot (s+2)}=\\
&=\frac1s\cdot \left(\frac As+\frac{B}{s+2}\right)=\\
&=\frac A{s^2}+\frac1s\cdot \frac B{s+2}=\\
&=\frac{A}{s^2}+\frac{C}{s}+\frac{D}{s+2}.\end{align}$$
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