Problem
ALG-B3-M07-P010 Argument with \(P(y)\)
#10
★★★☆☆ Level 3 of 5
Find all \(P\in\mathbb R[x]\) such that \(P(x+P(y))=P(x)+y\).
Compare degree in \(y\).
If \(d=\deg P\ge2\), then the degree of the left-hand side in \(y\) is \(d^2\), while the right-hand side has degree \(1\), impossible. Hence \(P(x)=ax+b\). Substitution gives \(a(x+ay+b)+b=ax+b+y\). Thus \(a^2=1\), \(ab=0\), so \(b=0\). The answer is \(P(x)=x\), \(P(x)=-x\).
Standard degree comparison in one variable.