Problem
ALG-B3-M07-P017 A Sum with a Product
#17
★★★★★ Level 5 of 5
Find all \(P\in\mathbb R[x]\) such that \(P(x+y)=P(x)+P(y)+2xy\).
Subtract \(x^2\).
Let \(Q(x)=P(x)-x^2\). Then \(Q(x+y)=P(x+y)-(x+y)^2=P(x)+P(y)+2xy-x^2-2xy-y^2=Q(x)+Q(y)\). Hence \(Q(x)=ax\). The answer is \(P(x)=x^2+ax\).
Polynomial correction to additivity.