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Polynomials with 3 real roots

Source: Poland Math Olympiad 1993 Round 1 #5

June 3, 2023

algebrapolynomial

Problem Statement

Prove that if the polynomial

x^3 + ax^2 + bx + c

has three distinct real roots, so does the polynomial

x^3 + ax^2 + \frac{1}{4}(a^2 + b)x + \frac{1}{8}(ab-c)