My polynomial is this ten term monster $P(x,y,z) = 6561 x^3+486 x^2+12 x+6561 y^3+1944 y^2+192 y+6561 z^3+6318 z^2+2028 z+223$
It's simplest form is ${1 \over 81} \left( (81x+2)^3 + (81y+8)^3 +(81z+26)^3 -33 \right)$
I'm tackling the problem of representing the number $33$ as the sum of three integer cubes, and if this polynomial has an integer root, then that will immediately supply the representation of $33$