Problem
ALG-B3-M07-P007 Multiplicative Polynomial
#7
★★★☆☆ Level 3 of 5
Find all \(P\in\mathbb R[x]\) such that \(P(xy)=P(x)P(y)\).
Do not forget constant solutions and nonzero roots.
The constant solutions are \(P=0\), \(P=1\). Let \(P\) be nonconstant. Then \(P(1)=1\). If \(r\ne0\) and \(P(r)=0\), then \(P(ry)=0\) for all \(y\), so \(P\equiv0\), a contradiction. Hence all roots, if any, are \(0\), and \(P(x)=ax^n\). Substitution gives \(a=a^2\), so \(a=1\). The answer is \(0\), \(1\), and \(x^n\), \(n\ge1\).
First serious root argument.