Polynomial and Complex Number

Let $\alpha$ be a root of polynomial
$$P(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1 x+a_0$$
where $a_i \in [0,1]$, for $i=1,2,\dots,n-1$. Prove that
$$Re(\alpha)<\frac{1+\sqrt{5}}{2}.$$

Proposed by Bogdan Enescu,Romania

Some Notes on Root of Unity

Let $a_,\dots,a_n$ be positive real numbers and let $\omega$ be a primitive nth root of unity. If the sides of an equiangular polygon have lengths $a_,\dots,a_n$ (in counterclockwise order) then
$$1+\omega+\omega^2+\dots+\omega^{n-1}=0$$;
$$a_1 + a_2 \omega +a_3 \omega ^2+ \dots + a_n \omega^{n-1}=0$$.