Problem
ALG-B3-M02-P012 Cubic extra term
#12
★★★★★ Level 5 of 5
Let \(f:\mathbb Q\to\mathbb Q\), \(f(x+y)=f(x)+f(y)+3xy(x+y)\), and \(f(1)=4\). Find \(f\).
Hint. Subtract \(x^3\).
Let \(g(x)=f(x)-x^3\). Since \((x+y)^3-x^3-y^3=3xy(x+y)\), the function \(g\) is additive. Also, \(g(1)=3\), so \(g(q)=3q\) on \(\mathbb Q\). Hence \(f(q)=q^3+3q\).
Goal: choose the right first substitution and then verify the found function.