If a+b+c=0 then prove that a3+b3+c3= 3abc , only when a=b=c

Dear student
It’s not required that a=b=c for this identity to hold true.
As we know
a3 +b3 + c3 – 3abc = (a+b+c) (a2 + b2 + c2 – ab – bc – ca)
If a+b+c = 0 , then your given statement is true
Also if a=b=c then either all are positive numbers or all should be negative numbers which cannot sum upto zero.

If a+b+c=0,Then a3+b3+c3= 3abc a+b+c=0 a+b= -c (a+b)3 = (-c)3 a3+b3+3ab (a+b) = -c3 a3+b3+3ab (-c)=-c3 a3+b3-3abc=-c3 a3+b3+c3=3abc Hence proved