Problem
ALG-B1-M04-P012 Common Root with the Third Iterate
#12
★★★★★ Level 5 of 5
Let \(P(x)\) be a monic quadratic trinomial. Suppose \(P(x)\) and \(P(P(P(x)))\) have a common root. Prove that \(P(0)P(1)=0\).
Inspired by final olympiad method · 2011 · Grade 9 · Problem 1
If \(r\) is a common root, then \(P(r)=0\). What is \(P(P(P(r)))\)?
Let \(r\) be a common root. Then \(P(r)=0\), so \(P(P(P(r)))=P(P(0))\). By the condition this is \(0\), hence \(P(P(0))=0\).
Write \(P(x)=x^2+ux+v\). Then \(P(0)=v\), and \(P(v)=0\) gives \(v^2+uv+v=v(v+u+1)=0\).
If \(v=0\), then \(P(0)=0\). If \(v+u+1=0\), then \(P(1)=1+u+v=0\). In both cases \(P(0)P(1)=0\).
Final-level method: the key is turning a common root into \(P(P(0))=0\).