If $N$ is large enough, then $x^5-Nx+1$ and $x^5-Nx^2+1$ are irreducible over $\mathbb{Q}$.

Show that if $N$ is large enough, then $x^5-Nx+1$ and $x^5-Nx^2+1$ are irreducible over $\mathbb{Q}$.

There is a hint for $x^5-Nx+1$: prove first that four of the roots in $\Bbb{C}$ has absolute values larger than $1$. I think that it follows from Rouche's theorem, but I don't know how to proceed.

from Hot Weekly Questions - Mathematics Stack Exchange