Equation on $\mathbb{R}$ : $(x+y+z)^3=x^3+y^3+z^3$

Question

How would i find all the $(x,y,z)\in \mathbb{R}$ verifying $(x+y+z)^3=x^3+y^3+z^3$ ?

Answer 1

You can rewrite the equation

$$(x+y+z)^3 - x^3 - y^3 - z^3 = 0$$

as

$$(x+y)(y+z)(x+z) =0$$

Making the solution identically those values such that at least one of the equations $$ x=-y\\ y=-z\\ z=-x $$ holds.