Knowing the zero coefficients in partial fractions from the beginning

Question

While solving partial fractions and getting the coefficients, can we know from the beginning that some coefficients will be zero? How?

For example $$\frac{1}{S^2(S^2+4)}=\frac{A}{S^2}+\frac{B}{S}+\frac{CS+D}{S^2+4}$$ If we compute the coefficients, we will get these values $$A=\frac{1}{4}$$ $$B=C=0$$ $$D=\frac{-1}{4}$$ Can we know before solving that B and C will be zeros? How?

Another example $$\frac{1}{(S^2+1)(S^2+4)}=\frac{AS+B}{S^2+1}+\frac{CS+D}{S^2+4}$$ $$B=\frac{1}{3}$$ $$A=C=0$$ $$D=\frac{-1}{3}$$ I am asking this question because while solving in a book, I found that the book assumed these coefficients to be zero from the beginning, i.e., it is written that $$\frac{1}{S^2(S^2+4)}=\frac{A}{S^2}+\frac{D}{S^2+4}$$ $$\frac{1}{(S^2+1)(S^2+4)}=\frac{B}{S^2+1}+\frac{D}{S^2+4}$$

Answer 1

For the time being, before partial fraction decomposition, let $x=S^2$ and everything will become clear.

After the partial fraction decomposition, just replace $x$ by $S^2$.

Answer 2

Since $\dfrac1{S^2(S^2+4)}$ is an even function, $B$ and $C$ must be $0$. The same argument applies to $\dfrac1{(S^2+1)(S^2+4)}$.