Problem
ALG-B3-M02-P017 Cubic normalization
#17
★★★★★ Level 5 of 5
Let \(f:\mathbb Q\to\mathbb Q\), \(f(x+y)-f(x)-f(y)=x^2y+xy^2\), and \(f(1)=1\). Find \(f\).
Hint. Compare the extra term with \((x+y)^3-x^3-y^3\).
Since \((x+y)^3-x^3-y^3=3x^2y+3xy^2\), put \(g(x)=f(x)-\frac{x^3}{3}\). Then \(g\) is additive. We have \(g(1)=1-\frac13=\frac23\), so on \(\mathbb Q\), \(g(q)=\frac{2q}{3}\). Answer: \(f(q)=\frac{q^3+2q}{3}\).
Goal: choose the right first substitution and then verify the found function.