Mathematics - Complex Number Question with Solution | TestHub

We have, $x=\omega -{\omega }^2-2$ or $x+2=\omega -{\omega }^2$ Squaring, $x^2+4x+4={\omega }^2+{\omega }^4-2{\omega }^3$ $={\omega }^2+{\omega }^3\omega \cdot -2{\omega }^3={\omega }^2+\omega -2\left[{\omega }^3=1\right]$ $=-1-2=-3\Rightarrow$ $x^2+4x+7=0$ Dividing $x^4+3x^3+2x^2-11x-6$ by $x^2+4x+7$ we get $x^4+3x^3+2x^2-11x-6=\left(x^2+4x+7\right)\left(x^2-x-1\right)+1$ $=(0)\left({\mathrm{x}}^2-\mathrm{x}-1\right)+1=0+1=1$