Problem

Source: 2017 Pan-African Shortlist - A6

Tags: algebra, polynomial, real root, inequalities



Let $n \geq 1$ be an integer, and $a_0, a_1, \dots, a_{n-1}$ be real numbers such that \[ 1 \geq a_{n-1} \geq a_{n-2} \geq \dots \geq a_1 \geq a_0 \geq 0. \]We assume that $\lambda$ is a real root of the polynomial \[ x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0. \]Prove that $|\lambda| \leq 1$.