Problem:

Let $\displaystyle P: \mathbb{C} \rightarrow \mathbb{C}, \hspace P(z)=z^n + \sum_{k=0}^{n-1}a_{k}z^{k}$ be a polynomial of degree $\displaystyle n\in\mathbb{N}$ with real coefficients $\displaystyle a_k \in [0,1]$ for $\displaystyle k=0,\dots,n-1.$

Show that if $\displaystyle P(z_0)=0$ for some $\displaystyle z_0 \in \mathbb{C}$ then $\displaystyle Re(z_0)<0$ or $\displaystyle |z_0|<(1+\sqrt{5})/2$.

Ideas :

I'm kind of stuck on this one.

I observed that there are clearly no roots on the positive real axis. I could use Descarte's Rule of Signs if necessary.

I also know, via the same method described

here, that all roots have modulus less than or equal to 2.

I was thinking of writing $\displaystyle z_0=r e^{i\theta}$ and using the statement that $\displaystyle Re(P(z_0))=0=Im(P(z_0))$ to get some relationship between $\displaystyle r$ and $\displaystyle \theta$ using trig functions but it hasn't panned out for me so far.

I also wonder if Vieta's formula for the coefficients in terms of the roots might help.