Mathematics - Complex Number Question with Solution | TestHub

We have $x^3+1\equiv (x+1)\left(x^2-x+1\right)$. Therefore, $\alpha$ and $\beta$ are the complex cube roots of -1 so that we may take $\alpha =-\omega$ and $\beta =-{\omega }^2,$ where $\omega \ne 1$ is a cube root of unity. Thus ${\alpha }^{100}=(-\omega {)}^{100}=\omega$ and ${\beta }^{100}={\left(-{\omega }^2\right)}^{100}={\omega }^2,$ so that the required equation is ${\mathrm{x}}^2+\mathrm{x}+1=0$