Problem
ALG-B3-M06-P007 A Square Correction
#7
★★★☆☆ Level 3 of 5
Find all \(f:\mathbb Z\to\mathbb Z\) such that \(f(m+n)=f(m)+f(n)+2mn\).
Subtract \(n^2\).
Let \(g(n)=f(n)-n^2\). Then \(g(m+n)=f(m+n)-(m+n)^2=f(m)+f(n)+2mn-m^2-2mn-n^2=g(m)+g(n)\). Hence \(g(n)=cn\), where \(c\in\mathbb Z\). The answer is \(f(n)=n^2+cn\).
Main technique of subtracting a polynomial part.