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P(x) = x^4 + a_1x^3 + a_2x^2 + a_3x + a_4 with a_i rational has 1 real root

Source: Austrian Polish 1981 APMC

April 26, 2020
polynomialrationalalgebra

Problem Statement

Let P(x)=x4+a1x3+a2x2+a3x+a4P(x) = x^4 + a_1x^3 + a_2x^2 + a_3x + a_4 be a polynomial with rational coefficients. Show that if P(x)P(x) has exactly one real root ξ\xi, then ξ\xi is a rational number.
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