Consider the following family of polynomials over $\mathbb{Q}$:
$$f_n = x^n - x^{n-1} - \dots - 1$$
Notice that these polynomials satisfy the recurrence
$$ f_{n+1} = x f_n - 1 $$
I would like to show that every polynomial in this family is irreducible over $\mathbb{Q}$. My strategy for showing this is to show that for each of these polynomials, all but one of the roots are in the unit circle (this would imply irreducibility since if $f_n(x) = g(x) h(x)$, then since $f_n$ has only one root of size greater than one, only one of $g, h$ can have non-zero constant term. This implies that 0 is a root of either $g$ or $h$, but 0 is not a root of $f_n$ for any $n$).
I am attempting to proceed by induction using Rouche's theorem. The idea is that if $f_n$ has $n-1$ roots in the unit circle, then $x f_n$ has $n$ roots in the unit circle. Therefore, if it's true that
$$ |f_{n+1} - x f_n | = 1 < |f_{n+1}| + |x f_{n}| $$
for all $x$ on the unit circle, then $f_{n+1}$ has $n$ roots inside the unit circle. However, I am not able to prove that we always have strict inequality above.
I suspect that this family $f_n$ has been studied before, but I was not able to find it in the literature. Any help would be appreciated.
