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f(x^2 + xy) = f(x)f(y) + yf(x) + xf(x+y) [Czech-Polish-Slovak Match 2018]

Source: Czech-Polish-Slovak Match 2018, Problem 1

July 2, 2018
functionfunctional equationalgebra

Problem Statement

Determine all functions f:Rā†’Rf : \mathbb R \to \mathbb R such that for all real numbers xx and yy, f(x2+xy)=f(x)f(y)+yf(x)+xf(x+y).f(x^2 + xy) = f(x)f(y) + yf(x) + xf(x+y). Proposed by Walther Janous, Austria
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