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bounding root of polynomial

Source: Bulgaria 1987 P1

June 15, 2021
algebraPolynomialsSequencespolynomial

Problem Statement

Let f(x)=xn+a1xn−1+…+an (n≥3)f(x)=x^n+a_1x^{n-1}+\ldots+a_n~(n\ge3) be a polynomial with real coefficients and nn real roots, such that an−1an>n+1\frac{a_{n-1}}{a_n}>n+1. Prove that if an−2=0a_{n-2}=0, then at least one root of f(x)f(x) lies in the open interval (−12,1n+1)\left(-\frac12,\frac1{n+1}\right).
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