Question

if x+y+z=0 what is the value of
${x}^{3} + {y}^{3} + {z}^{3}$

Answer 1

I don't think that there's a unique answer, because it depends on the numbers you choose: for example, if you pick $x = -1,\ y=0,\ z= 1$, then you have

$x+y+z=0,\quad x^3+y^3+z^3 = 0$

But for example, if you if you pick $x = -2,\ y=1,\ z= 1$, then you have

$x+y+z=0,\quad x^3+y^3+z^3 = -8+1+1 = -6$

Even if you wanted to use formula, you can at most solve one variable in terms of the other two:

$x+y+z \iff x=-y-z \iff x^3 = -y^3 - 3 y^2 z - 3 y z^2 - z^3$

and thus

$x^3+y^3+z^3 = -y^3 - 3 y^2 z - 3 y z^2 - z^3+y^3+z^3 = - 3 y^2 z - 3 y z^2$

Answer 2

$x {}^{3} + y {}^{3} + z {}^{3} = (x + y + z)(x {}^{2} + y {}^{2} + z {}^{2} - xy - yz - xz) + 3xyz$
if we put the value of x+y+z in the equation so the 3xyz will remain so the answer is 3xyz.