Question

if x+y+z=0 show thatx^3+y^3+z^3=3xyz
z

Answer 1

We know the formula,

(a+b+c)^3 =

Refer to pic I'm sending

So if

x+y+z = 0

(x+y+z)^3 = x^3+y^3+z^3 - 3xyz

0 + 3xyz = x^3+y^3+z^3

Answer 2

Given, x3 + y3 + z3 = 3xyz

Therefore, x3 + y3 + z3 - 3xyz =0

This means,

x3 + y3 + z3 - 3xyz = (x + y + z) (x2 + y2 + z2 - xy - yz - zx)

Now if x + y + z = 0, then,

x3 + y3 + z3 - 3xyz = ( 0 ) (x2 + y2 + z2 - xy - yz - zx) 

[ subsituting the value of x + y + z ]

x3 + y3 + z3 - 3xyz = 0   

x3 + y3 + z3 = 3xyz .