Problem
ALG-B2-M06-P002 Cubes through p, q, r
#2
★★☆☆☆ Level 2 of 5
Let \(p=a+b+c\), \(q=ab+bc+ca\), \(r=abc\). Prove \[a^3+b^3+c^3=p^3-3pq+3r.\]
Hint. Use \(a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)\).
By the identity, \(\sum a^3-3r=p((p^2-2q)-q)=p(p^2-3q)=p^3-3pq\). Therefore \(\sum a^3=p^3-3pq+3r\).
This identity will be used repeatedly in Schur-type problems.