I want to find the root of a function $f$ defined as
$$ f(x)= e^{-cx} - \frac{P_n(x)}{Q_m(x)}$$
where $x,c$ are real numbers and $P_n,Q_m$ are irreducible polynomials of rank $n$ and $m$ respectively, whose coefficients depend on $c$. If I understand the current state of the art, a closed-form solution for $x$ can be expressed in terms of the generalised Lambert W only if $P$ and $Q$ are reducible, i.e. can be factorised to $P_n=\prod_{i=0}^n(x-t_i)$ and $Q_m=\prod_{i=0}^m(x-t_i)$. Is this really the case or am I misunderstanding the literature (I am not a mathematician)? In the literature I do not see an explicit discussion of this issue with irreducible polynomials.
So far I based myself on:
$$e^{-cx}-\frac{P_n(x)}{Q_m(x)}=0$$ $$-\frac{P_n(x)}{Q_m(x)}=-e^{-cx}$$ $$\frac{P_n(x)}{Q_m(x)}=e^{-cx}$$ $$\frac{P_n(x)}{Q_m(x)}e^{cx}=1$$
Each polynomial with complex coefficients can be factorized over the complex numbers. This is stated by the fundamental theorem of algebra. Now you can apply Generalized Lambert W.