<s>[INST] An uncertainty for $a^3+b^3+c^3-3abc$ I have a doubt regarding $a^3+b^3+c^3-3abc$.
For factoring, it is easy that if $a+b+c=0$, then $a^3+b^3+c^3=3abc$ as $a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$
But on using AM-GM inequality, we see-
$$ {a^3+b^3+c^3\over 3}\geq abc \Rightarrow a^3+b^3+c^3\ge3abc $$
AM-GM inequality ensures that equality holds if and only if all variables are equal.
So, equality holds if and only if $a=b=c$, which is trivial.
But we see by factoring that equality holds also if $a+b+c=0$. As the inequality is nothing more than a mere AM-GM, so equality should hold where the inequality ensures us it holds. But it also holds for $a+b+c=0$.
How is it possible and if it is, how can I find such equality cases?
 [/INST] When $a+b+c=0$, then $a,b,c$ all of $a,b,c$ are not  positive, So AM_GM will not be used.
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