Question

if a + b + C is equal to zero prove that a cube plus b cube + C cube is equal to 3 ABC

Answer 1

There is a formula
a cube + b cube +c cube -3abc=(a+b+c)(a sq + b sq +c sq + 2 ab + 2 bc
+2 ca)
given a + b+c =0
so right hand side is totally zero then
a cube + b cube + c cube = 3abc
proved

Answer 2

Given, a + b + c = 0 can be written as

            a + b = -c   ---------------- (1)

n cubing both sides, we get

(a+b)^3 = (-c)^3

a^3 + b^3 + 3ab(a+b) = -c^3

a^3 + b^3 + c^3 = -3ab(a+b)

a^3 + b^3 + c^3 = -3ab(-c) (from (1))

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


Hope this helps!