Problem
ALG-B3-M11-P008 Product of Arguments
#8
★★★★★ Level 5 of 5
Find all \(P\in\mathbb R[x]\) if \(P(xy)=P(x)P(y)\).
Handle constant solutions and nonzero roots.
The constant solutions are \(0\), \(1\). If \(P\) is nonconstant and has a nonzero root \(r\), then \(P(ry)=0\) for all \(y\), impossible. Hence \(P(x)=ax^n\). Substitution gives \(a=a^2\), so \(a=1\). The answer is \(0\), \(1\), and \(x^n\), \(n\ge1\).
Polynomial root argument.