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