If the polynomial is $a_nz^n+a_{n-1}z^{n-1}+\ldots+a_0$, then if $x$ is a root, then $(|x|-1)\cdot(|x|^{n-1}+|x|^{n-2}+\ldots+|x|+1)=|x|^n-1<|x|^n$ $\leq|a_nx^n|=|-a_{n-1}x_{n-1}-\ldots-a_0|\leq|a_{n-1}| |x|^{n-1}+\ldots+|a_1| |x|+|a_0|$ so some $|a_i|$ is greater than $|x|-1$, and hence $|a_i|\geq |x|$.