So say I have a partial fraction with an arbitrary numerator as follows:
$$\frac{f(s)}{(s-s_1)\cdots(s-s_k)}$$ $$s_i\neq s_j, \forall i \neq j$$
With regards to decomposing it into the sum of fractions, can one always treat it as the following:
$$\frac{f(s)}{(s-s_1)\cdots(s-s_k)}=f(s)\frac{1}{(s-s_1)\cdots(s-s_k)} = f(s) \left( \frac{A_1}{(s-s_1)}+\cdots+\frac{A_k}{(s-s_k)} \right)$$
Or is there some special rule that can forbid this? (for example I know that if the top is a polynomial of degree less than the denominator you do not factor it out)
For anyone who is interested in more context, this question came to mind while working with Laplace transforms where I came across a similar example with $f(s)=e^{-s}$ where I know that it can be done but I do not have a solid enough understanding to know whether it can always be done or not