Consider $$ P(X) = X^{n+1} - aX^{n} +aX -1 $$ with $a \in [-1;1]$.
Show that the moduli of the roots of $P$ are $1$.
What I tried is
I've noticed if $z$ is a root then $\frac{1}{z}$ is a root as well.
I've checked:
- $X-a+aX-1 = (1+a)(X-1)$
- $X^{2} - aX -1 +aX = (X-1)(X+1)$
- $X^{3} - aX^{2} +aX-1$ is more subtle.
I don't how far this question could lead us but I know compelex analysis and basic algebra.