@GeraldEdgar: do you have a source (perhaps with proof) where I can look up these formulas?

Thanks for the comments! It seems to me that the formulas in mathoverflow.net/questions/186303/… don't work for a=1. Since they run from $-\infty$ to $+\infty$ they include n=0 and for a=1 the summand is not defined.

Thanks, Joachim. Nice idea. I am just interested in exactly what you were saying: for concrete examples of say families of polynomials, did anybody try to make the "idea" in my question work?

Yes, thanks, that's the question. $f$ is reducible.

Thanks for setting me on the right track! I found chapter 3 of Antonio Cafure, & Eda Cesaratto. (2017). Irreducibility Criteria for Reciprocal Polynomials and Applications. The American Mathematical Monthly, 124(1), 37-53 quite helpful. The functions $f_k$ there are more or less Chebyshev polynomials.

Thanks a lot! Nice and simple idea.