Question

Let $\alpha$ be a root of the equation $1+x^{2}+x^{4}=0$. Then the value of $\alpha^{1011}+\alpha^{2022}-\alpha^{3033}$ is equal to:

• A. 1
• B. $\alpha$
• C. $1+\alpha$
• D. $1+2 \alpha$

Answer 1

Answer: (A)

$x^{4}+x^{2}+1=0$
$\Rightarrow\left(x^{2}+x+1\right)\left(x^{2}-x+1\right)=0$
$\Rightarrow x=\pm \omega, \pm \omega^{2}$ where $\omega=1^{1 / 3}$ and imaginary.
So $\alpha^{1011}+\alpha^{2022}-\alpha^{3033}=1+1-1=1$