Let α be a root of the equation 1 + x ^ 2 + x ^ 4 = 0 . Then the...

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$</p><p>$⇒ x = \pm \omega, \pm {\omega}^{2}$where$\omega = {1}^{1 / 3} and imaginary.

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