Simplify homogeneous polynomial of degree 3 with all coefficients equal 1

Question

I want to simplify this homogeneous polynomials: $$x^3 + y^3 + z^3 + x^2 y + x y^2 + x^2 z + x z^2 + z^2 y + z y^2 + x y z$$ to a simpler form. For example, the homogeneous polynomials of degree 2 can be simplified as follows: $$x^2 + y^2 + z^2 + x y + x z + y z = \frac{1}{2}((x + y)^2 + (x + z)^2 + (y + z)^2)$$. So I wonder whether degree of 3 could also be simplified. I know this kind of "simplification" is not very well-defined and very dependent on experience. I have tried mathematica but in vain. Is there any simpler form?

Answer 1

Maybe $$(x^2+y^2+z^2)(x+y+z)+xyz$$ or $$\frac{x^5y+y^5z+z^5x-x^5z-y^5x-z^5y}{(x-y)(x-z)(y-z)},$$ where $\prod\limits_{cyc}(x-y)\neq0?$