<s>[INST] $(x + y + z)....(x + y\omega_n^{n-1} + z\omega_n) = x^n + y^n + z^n - P(x,y,z)$ To find $P$ $$(x + y + z)(x + y\omega_n + z\omega_n^{n-1})(x + y\omega_n^2 + z\omega_n^{n-2})....(x + y\omega_n^{n-1} + z\omega_n) = x^n + y^n + z^n - P(x,y,z)$$ where $\omega_n$ is an nth root of unity.
The question is to find the polynomial $P$.
I have tried to manually multiply the terms of LHS and then equate the coefficients to get the polynomial but that's too cumbersome:
$(x + y + z)(x + y\omega_n + z\omega_n^{n-1})(x + y\omega_n^2 + z\omega_n^{n-2})....(x + y\omega_n^{n-1} + z\omega_n) = x^n(1 + [Y + Z])(1 + [Y\omega_n + Z\omega_n^{n-1}])(1 + [Y\omega_n^2 + Z\omega_n^{n-2}])....(1 + [Y\omega_n^{n-1} + Z\omega_n])$ where $Y=\frac {y}{x}$ and $Z=\frac {z}{x}$ 
Hence, we can apply the formula:
$(1+\alpha)(1+\beta)(1+\gamma)...... = 1 + [\alpha + \beta + \gamma + ...] + [\alpha\beta + \beta\gamma + ....] + ....$
I hope someone can help.
 [/INST] As suggested in the first comment, computationally it looks like $$P\equiv P_n=\dfrac{x^n}{t^n}L_n(t)-x^n=\dfrac{x^n}{t^n}(L_n(t)-t^n), $$
where $L_n$ is the $n$th Lucas polynomial in $t:=\dfrac{{ix}}{\sqrt{yz^{\phantom l}}}$.
E.g. for $n=6$, $$P=6x^4yz-9x^2y^2z^2+2xy^3z^3$$ while $$L_6(t)=t^6+6t^4+9t^2+2.$$
</s>