Question

If $\alpha$ is a complex number such that ${\alpha }^2-\alpha +1=0$, then ${\alpha }^{2011}=$

• A. $-\alpha$
• B. ${\alpha }^2$
• C. $\alpha$
• D. $1$

Answer 1

Answer: (C)

$\alpha$

Since ${\alpha }^3+1=(\alpha +1)({\alpha }^2-\alpha +1)$, using the given information, we get ${\alpha }^3+1=0$

$\Rightarrow {\alpha }^3=-1$

$\Rightarrow {\alpha }^{2011}={\alpha }^{2010}\times \alpha$

$\Rightarrow {\alpha }^{2011}=({\alpha }^3{)}^{670}\times \alpha$

$\Rightarrow {\alpha }^{2011}=(-1{)}^{670}\times \alpha$

$\Rightarrow {\alpha }^{2011}=\alpha$