If $\alpha$ is a root of $p(x)$ where $\alpha \in \mathbb{C}$, show that $\vert \alpha \vert \leq \{1, \sum_{i=0}^{n-1} \vert a_i \vert \}$

Question

If $\alpha$ is a root of $p(x)= x^n+a_{n-1}x^{n-1}+\cdots a_0$ where $\alpha \in \mathbb{C}$, show that $$\vert \alpha \vert \leq \{1, \vert a_0\vert +\vert a_1 \vert + \cdots + \vert a_n\vert\}$$

Answer 1

If $|α| \leqslant 1$, it is done. If $|α| > 1$, then$$ |α|^n = |α^n| = \left| \sum_{k = 0}^{n - 1} a_k α^k \right| \leqslant \sum_{k = 0}^{n - 1} |a_k| |α|^k \leqslant |a|^{n - 1} \sum_{k = 0}^{n - 1} |a_k|, $$ thus$$ |α| \leqslant \sum_{k = 0}^{n - 1} |a_k|. $$