Is it possible to factor $x^3+y^3+z^3$

Question

It's a quick question: is it possible to factor $$x^3+y^3+z^3$$ into a form like $(x+y+z)(\ldots)$?

Answer 1

As an alternative to the other answer, note that $(x + y + z)(...)$ has a zero at $2,2,-4$, while $2^3 + 2^3 + (-4)^3 = -48$.

Answer 2

The polynomial $x^3+y^3+z^3$ is irreducible in $\mathbf{C}[x,y,z]$. More generally, so is $x^n+y^n+z^n$ for $n \geq 1$. See the answers given to this question.

Answer 3

General factorization over polynomials is not possible. But factorization is possible in special cases under Diophantine solution i.e. if $x,y$ and $z$ are positive integers such that $3xyz$ is divisible by $x+y+z$ then factorization is possible.

It is possible to creeate infinitely many such Diophantine solution in parametric form. For example $x = p, y = 2p, z = 3p$.