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P(X)^2-Q(X)^2 has a rational root

Source: Romanian TST 1997

September 17, 2011
algebrapolynomialinequalitiesalgebra proposed

Problem Statement

Let P(X),Q(X)P(X),Q(X) be monic irreducible polynomials with rational coefficients. suppose that P(X)P(X) and Q(X)Q(X) have roots α\alpha and β\beta respectively, such that α+β\alpha + \beta is rational. Prove that P(X)2−Q(X)2P(X)^2-Q(X)^2 has a rational root.
Bogdan Enescu
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