Let $k\geq 3$ and $p\geq 1$ be integers. Set $f_{k,p}(x):=x^k-px^{k-1}-(p+1)x-1$. I need to prove for which parameters $k$ and $p$, this polynomial is irreducible (over $\mathbb{Z}$). For instance, $f_{5,1}(x)$ is the only example that I find to be reducible, since it is divisible by $x^2-x-1$.
Anyone could give some suggestion or idea? Some useful criterion (I was not able to use Eisenstein and Perron method also does not work).
Thanks in advance.