MathDB

Problem 1

Part of 1987 Bulgaria National Olympiad

Problems(1)

bounding root of polynomial

Source: Bulgaria 1987 P1

6/15/2021
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).
algebraPolynomialsSequencespolynomial