While doing work on another problem, I came across the following family of polynomials: $$P_e(x) := ex^{2e} - x^{2e-1} - x^{2e-2} - \cdots - x + e\in \mathbb{Z}[x],$$ where $e\geq 1$ is an integer. After doing some computations in sage, it seems that the polynomials $P_e$ are irreducible; this I checked for all $e\leq 500$. On the other hand, I don't see a simple argument why this should be the case.
Question: Are the polynomials $P_e$ always irreducible for $e\geq 1$?
Of course one natural thing to try is to find a prime number $q$ such that $P_e$ is irreducible mod $q$. There is no guarantee this would work, since there could be $P_e$ that are irreducible, yet reducible modulo every prime $q$. And indeed, this appears to be the case with $P_{12}$ (and some others); at least sage tells me $P_{12}$ is reducible modulo every prime $q<60000$.
I also see that the polynomials $P_e$ are self-reciprocal. Perhaps this added structure could be useful in checking for irreducibility, though I'm not sure how.
Any suggestions or insights are welcome, as this is really not my area.