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P(a)P(b)=-(a-b)^2 => Prove that P(a)+P(b)=0

Source: Iran Third Round MO 1997, Exam 2, P1

June 30, 2012
algebrapolynomialalgebra proposed

Problem Statement

Let PP be a polynomial with integer coefficients. There exist integers aa and bb such that P(a)⋅P(b)=−(a−b)2P(a) \cdot P(b)=-(a-b)^2. Prove that P(a)+P(b)=0P(a)+P(b)=0.
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