Roots and coefficients

Setup. If $\alpha$ is a root, $P(\alpha)=0$, so
$$\alpha^n=-(a_{n-1}\alpha^{n-1}+\cdots+a_0).$$

Bound. Let $M=\max_j|a_j|$. If $|\alpha|>1$:
$$|\alpha|^n\le M(|\alpha|^{n-1}+\cdots+1)=M\frac{|\alpha|^n-1}{|\alpha|-1}\le M\frac{|\alpha|^n}{|\alpha|-1}.$$

Conclude. Dividing by $|\alpha|^n$: $1\le M/(|\alpha|-1)$, i.e. $|\alpha|\le M+1$.

Conclusion. All roots satisfy $|\alpha|\le1+M$ with $M=\max|a_j|$.