TITLE: Show that polynomial $X^{n+1}-aX^n+aX-1$ has only roots of module $1$
QUESTION [9 upvotes]: I gave an exercise to a student the other day, thinking I had a simple solution for it, but it seems that my solution was just bullshit (at least, not conclusive). Well, at least, I warned her I wasn't sure of my solution :-)
Here it is : $a\in[-1,1]$, show that all roots of polynomial $P=X^{n+1}-aX^n+aX-1$ have modulus $1$ (I try different ways of expressing the problem, tell me which one is better english...).
It is easy to see that 

roots of $P$ are $1$, $-1$ for some special cases, and complex not real roots $z_i$ such that $\overline{z_i}$ is also a root,
as $P$ is kind of reciprocal, $z$ root implies $\frac1z$ is also a root$.

All the other things I tried are not conclusive. For example, $a\in[-1,1]$ may be translated as $a=\cos\theta$ for some $\theta\in\mathbb R$, but I don't see how to use this. I tried some rewriting, but nothing seems to work. I tried to work on the modulus of a root, tried the relations roots-coefficients...
I'm quite stuck here. Could you help me please ?
Thanks.

REPLY [5 votes]: The equation can be rewritten as
$$
B(x)=x^n\frac{x-a}{1-ax}=1.
$$
The function $B$ is analytical in the unit disc for $a\in(-1,1)$ (the cases $a=\pm 1$ are simple), and $|B(x)|=1$ on the unit circle. Now apply the maximum modulus principle to $B(x)$ and $B(1/x)$ to ensure that the equality is only possible on the unit circle.