Important Algebra concept frequently touched

[Ravisankar Vemuri]

Various types of questions are framed basing on this concept: 

a³+b³+c³-3abc = 0

=> (a+b+c)(a²+b²+c²-ab-bc-ca) = 0

=> a+b+c = 0 (or) a²+b²+c²-ab-bc-ca = 0

a²+b²+c²-ab-bc-ca = 0 => 2(a²+b²+c²-ab-bc-ca) = 0

=> (a-b)²+(b-c)²+(c-a)² = 0 => a-b = b-c = c-a = 0 => a=b=c

If a³+b³+c³-3abc = 0, then the possibilities are

a+b+c = 0 (2) a²+b²+c²= ab+bc+ca (3) a=b=c