I have a fractional function $\frac{1}{x(1+x^{n-1})}$. using PFD: $\frac{A}{(1+x^{n-1})}+ \frac{B}{x}$, that means $Ax+(1+x^{n-1})B=1$.
For this to hold, we need $A=0, B=0, B=1$, which is of course impossible.
Does that mean that this fraction cannot be decomposed? I remember reading that all fractions of polynomials can be decomposed.
A and B can be polynomials.
In this case, $A=-x^{n-2},B=1$ Works.