Problem
ALG-B1-M02-P011 Divisibility of a Cubic Expression
#11
★★☆☆☆ Level 2 of 5
Prove that for integers \(a,b,c\), the expression \(a^3+b^3+c^3-3abc\) is divisible by \(a+b+c\).
Use the cubic identity from the theory section.
By the identity, \(a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)\).
The second factor is an integer when \(a,b,c\) are integers. Therefore the expression is divisible by \(a+b+c\).
The key point is that divisibility follows from exact factorisation.