$\mathbf {The \ Problem \ is}:$ If $z \in \mathbb C$ is a root of the monic polynomial $t^n+a_1t^{n-1}+....+a_n \in \mathbb C[t]$ , then show that $| z |\leq max\{ 1, |a_1|+|a_2|+...+|a_n|\} .$
$\mathbf {My \ approach} :$ Actually, I did think of the fact that when $| z| \geq 1$ , then I can construct a function including a complex root $z$ such that we can get the maximum value of $| z|$ from there.
Another thought came into my mind, i.e. by using the $\mathbf {Viete's}$ formula about adding all possible permutations of roots , then the upper bound other than $1$ is obtained on right side by triangular in-equality .
A small hint is warmly appreciated.
Suppose $z^{n}+a_1z^{n-1}+...+a_n=0$. If $|z| \leq 1$ then the conclusion is trivially true. Suppose $|z| >1$. Then $|z|^{n}=|(a_1z^{n-1}+...+a_n)|\leq |z|^{n-1} (|a_1|+|a_3|+...|a_n|)$. Divide by $|z|^{n-1}$.