If $a,b,c \in R$ are distinct, then $-a^3-b^3-c^3+3abc \neq 0$. If $a,b,c \in R$ are distinct, then $-a^3-b^3-c^3+3abc \neq 0$.
I think it is trivial because they are distinct.
So I wonder just saying "Since they are distinct" is enough to prove it?
Of course there could be several more detailed versions but I just want to know that reasoning is true or not.

You statement is wrong, see $(0,-1,1)$.
