Partial fractions zero coefficients

Question

In partial fractions, there can be terms in the coefficients in the partial form that turn out to be zero. One case of this is when the variable is "linear" in a power of $x$, for example, found in this similar question $\frac{1}{x^2(x^2+4)}=\frac{A}{x^2}+\frac{B}{x}+\frac{Cx+D}{x^2+4}$, and because of this, we intuitively know C is zero. This is sufficient in this case, but not necessary. For example, $\frac{3x+3}{(x^2+x+2)(x^2+4x+5)}=\frac{Ax+B}{x^2+x+2}+\frac{Cx+D}{x^2+4x+5}$. Solving, $A=0\ B=1\ C=0\ D=-1$. Is there, either a necessary condition in general or other sufficient conditions?

Answer 1

If the function $ f(x) $ to decompose can be written as $ F(x^2) $, then the decomposition will contain only terms with $ x^2$. If not, we can say nothing. It depends on the expression of $ f(x)$.

This is true for your first example but not for the second.