Given $p(x)$ is a polynomial of degree $n$, and $r$ to be its root, how to prove that $|r| \le \max(1, \sum_{i=1}^n |{a_i \over a_0}|)$? Let $p(x)=a_0x^n+a_1x^{n-1}+\cdots+a_n,a_0 \ne 0$ to be univariate (1 variable) polynomial of degree $n$.
Let $r$ be its root, i.e. $p(r)=0$.
How can I prove that:
$$|r| \le \max\left(1, \sum_{i=1}^n \left|{a_i \over a_0}\right|\right)$$
And is this always true?
$$|r| \le \sum_{i=1}^n \left|{a_i \over a_0}\right|$$

Hint: Suppose that $|r|\ge 1$. Then for any $k$, $|a_kr^k|\le |a_kr^{n-1}|$. 
It follows (Triangle Inequality) that 
$$|a_1r^{n-1}+a_2r^{n-2}+\cdots +a_n|\le |r^{n-1}|(|a_1|+|a_2|+\cdots +|a_n|).$$
But if $a_0r^n+a_1r^{n-1}+\cdots+a_n=0$, then
$|a_0r_n|=|a_1r^{n-1}+a_2r^{n-2}+\cdots +a_n|$. 
