Problem
ALG-B3-M01-P019 Cubic extra term
#19
★★★★★ Level 5 of 5
Let \(f:\mathbb Q\to\mathbb Q\), \(f(x+y)=f(x)+f(y)+3xy(x+y)\), and \(f(1)=1\). Find \(f\).
Hint. Use \((x+y)^3-x^3-y^3=3xy(x+y)\).
Put \(g(x)=f(x)-x^3\). Then \(g(x+y)=g(x)+g(y)\), because the cubic extra term is completely removed. Also, \(g(1)=0\), so on \(\mathbb Q\), \(g(q)=0\). Therefore \(f(q)=q^3\). Verification follows from the identity above.
Goal: separate guessing the answer from a complete proof and domain check.