Magnitude of a root of a polynomial

Let $P(X):=\sum_{i=0}^{n}a_{i}X^{i}$ be a non-constant polynomial with complex coefficients.

It is necessarily the case that any of its roots $\lambda$ verify $|\lambda| \leqslant \sum_{i=0}^n\left|a_i\right|$ ?

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Solution by Sequoia

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Assume the inequality holds for any root of any complex polynomial of degree at least $1$ .

Then any root $\lambda$ of $P(X):=\sum_{i=0}^{n}a_{i}X^{i}$ is also a root of $\alpha P$ where $\alpha$ is a non-nul complex number. Hence if $\lambda$ is a solution of the inequality for $P$ then it has to be a solution for $\alpha P$ . Hence, for all $\alpha\in\mathbb{C}^{*}$ :
$\vert\lambda\vert \leqslant \vert\alpha\vert\,\sum_{i=0}^{n}\vert a_{i}\vert$ But then one can let $\alpha$ goes to $0$ so $\lambda$ has to be zero. But there exists polynomials which roots are not zero thus this is our contradiction.

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