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Sum of polynomials is not equal to x^2+7

Source: LXII Polish Olympiad 2011, Problem 6

May 24, 2011
algebrapolynomialalgebra unsolved

Problem Statement

Prove that it is impossible for polynomials f1(x),f2(x),f3(x),f4(x)∈Q[x]f_1(x),f_2(x),f_3(x),f_4(x)\in \mathbb{Q}[x] to satisfy f12(x)+f22(x)+f32(x)+f42(x)=x2+7.f_1^2(x)+f_2^2(x)+f_3^2(x)+f_4^2(x) = x^2+7.
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