Mathematics - Quadratic Equation Question with Solution | TestHub

Given, $x^4+x^2+1=0$

Using the formula $a^4+a^2+1=\left(a^2+a+1\right)\left(a^2-a+1\right)$

We get, $\left(x^2+x+1\right)\left(x^2-x+1\right)=0$

$\Rightarrow x=\pm \omega , \pm {\omega }^2$ where $\omega =1^{1/3}$ and imaginary.

So ${\alpha }^{1011}+{\alpha }^{2022}-{\alpha }^{3033}=1+1-1=1$