I want to write
P(x,y,z)=x^{3}y + x^{3}z + y^{3}x + y^{3}z +z^{3}x + z^{3}y
in terms of elementary symmetric polynomials, but I'm getting stuck at the first step.

I know the general procedure is to pick the largest monomial (lexicographically) and subtract it's elementary symmetric 'form' from both sides. But I can't figure out how to write z^{3}y, the biggest monomial, like this.