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Bound on real root of polynomial with non-increasing coefficients

Source: 2017 Pan-African Shortlist - A6

May 5, 2019
algebrapolynomialreal rootinequalities

Problem Statement

Let n≥1n \geq 1 be an integer, and a0,a1,…,an−1a_0, a_1, \dots, a_{n-1} be real numbers such that 1≥an−1≥an−2≥⋯≥a1≥a0≥0. 1 \geq a_{n-1} \geq a_{n-2} \geq \dots \geq a_1 \geq a_0 \geq 0. We assume that λ\lambda is a real root of the polynomial xn+an−1xn−1+⋯+a1x+a0. x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0. Prove that ∣λ∣≤1|\lambda| \leq 1.
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