a+b+c=0a^3+b^3+c^3=?

We use the identity that

a^3 + b^3 + c^3 - 3abc = (a+b+c)(a^2 + b^2 + c^2 - ab - bc - ca)

Now, if a+b+c = 0, the RHS becomes zero, thus the LHS is also zero

=> a^3 + b^3 + c^3 - 3abc = 0

=> a^3 + b^3 + c^3 = 3abc