This is from my textbook. I am having trouble working out the calculations that the author skips over.
So we start with the polynomial $\ X^3 - aX^2 + bX -c$ with zeros $x_1,x_2,x_3$. Then we define $u_1=x_1 + \omega x_2 + \omega^2 x_3$ and $u_2=x_1+\omega^2 x_2 +\omega x_3$. We show that $u_1 u_2$ and $(u_1)^3 + (u_2)^3$ are symmetric polynomials so can be expressed in terms of $a,b,c$. The next line is where I got stuck.
$$(u_1)^3 + (u_2)^3 = 2(x_1^3+x_2^3+x_3^3) - 3(x_1^2x_2 +x_1x_2^2+x_1^2x_3+x_1x_3^2+x_2^2x_3+x_2x_3^2)$$
When I multiplied out $(u_1)^3+(u_2)^3$, I got the following: $$2(x_1^3+x_2^3+x_3^3) +(3\omega + 3\omega^2)(x_1^2x_2+x_1x_2^2+x_1^2x_3+x_1x_3^2+x_2^2x_3+x_2x_3^2) + 12x_1x_2x_3$$
I'm not sure if there is a way I can simplify $3\omega + 3\omega^2$ and I'm also confused why the $x_1x_2x_3$ term goes away.