Problem

Source: 2024 CTST P10

Tags: algebra, polynomial, 2024 CTST



Let $M$ be a positive integer. $f(x):=x^3+ax^2+bx+c\in\mathbb Z[x]$ satisfy $|a|,|b|,|c|\le M.$ $x_1,x_2$ are different roots of $f.$ Prove that $$|x_1-x_2|>\frac 1{M^2+3M+1}.$$ Created by Jingjun Han