Question

if $\alpha ,\beta$ are the roots of the equation $x^2-x+1=0$, then ${\alpha }^{2009}+{\beta }^{2009}$ =

• A. -1
• B. 1
• C. 2
• D. -2

Answer 1

The correct option is B 1

$x^2-x+1=0\Rightarrow x=\frac{1\pm \sqrt{1-4}}{2}x=\frac{1\pm \sqrt{3}i}{2}\alpha =\frac{1}{2}+i\frac{\sqrt{3}}{2}=-{\omega }^2\beta =\frac{1}{2}-\frac{i\sqrt{3}}{2}=-\omega {\alpha }^{2009}+{\beta }^{2009}=(-{\omega }^2{)}^{2009}+(-\omega {)}^{2009}=-{\omega }^2-\omega =1$