Unity Application

Clearly $\alpha$ , $\beta$ and $\gamma$ are the cubic roots of unity. We know that $\alpha=1$ , $\beta=e^\dfrac{2\pi i}{3}$ and $\gamma=e^{-\dfrac{2\pi i}{3}}$ . So,

$x=e^\dfrac{2\pi i}{3}-e^{-\dfrac{2\pi i}{3}}-1-1 \\ x=2i \sin \left(\dfrac{2 \pi}{3}\right)-2 \\ x=\sqrt{3}i-2$

Manipulate that expression a little:

$x+2=\sqrt{3}i \\ (x+2)^2=(\sqrt{3}i)^2 \\ x^2+4x+4=-3 \\ x^2+4x+7=0$

Now, using long polynomial division we know that:

$x^4+3x^3+2x^2-11x-6=(x^2+4x+7)(x^2-x-1)+1 \\ x^4+3x^3+2x^2-11x-6=(0)(x^2-x-1)+1 \\ x^4+3x^3+2x^2-11x-6=\boxed{1}$

(\beta,\gema=\dfrac{-1\pm\sqrt3} 2. \alpha=1.\
\therefore\ x=-2+\sqrt3i.\
x= - 2 + \sqrt3i.\
x^2= 1 - 4\sqrt3i.\
x^3= 10 + 9\sqrt3i.\
x^4= - 47 - 8\sqrt3i.\
\therefore x^4+3x^3+2x^2 - 11x - 6=1.