## Roots of a Polynomial

1. The problem statement, all variables and given/known data

Let $z^n + \sum_{k=0}^{n-1}a_kz^k$ be a polynomial with real coefficients $a_k\in[0,1]$. If $z_0$ is a root, prove that $Re(z_0) < 0$ or $|z_0| < \frac{1+\sqrt{5}}{2}$.

2. Relevant equations

3. The attempt at a solution

I have attempted to solve this problem by contradiction (i.e. assuming there is a root $z_0$ with $Re(z_0) \geq 0$ and $|z_0| \geq \frac{1+\sqrt{5}}{2}$). I then tried to look for a contradiction in the equations $Re(z_0^n + \sum_{k=0}^{n-1}a_kz_0^k) = 0$ and $Im(z_0^n + \sum_{k=0}^{n-1}a_kz_0^k) = 0$. Unfortunately, I'm not able to find any contradiction.
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