Question

If $\alpha$ and $\beta$ 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

Answer: (B)

$x^2-x+1=0\Rightarrow x=\frac{1\pm \sqrt{1-4}}{2}$
$x=\frac{1\pm \sqrt{3}i}{2}$
$\alpha =\frac{1}{2}+i\frac{\sqrt{3}}{2},\beta =\frac{1}{2}-\frac{i\sqrt{3}}{2}$
$\alpha =cos\frac{\pi }{3}+isin\frac{\pi }{3},\beta =cos\frac{\pi }{3}-isin\frac{\pi }{3}$
${\alpha }^{2009}+{\beta }^{2009}=2cos2009\left(\frac{\pi }{3}\right)$
$=2cos\left[668\pi +\pi +\frac{2\pi }{3}\right]=2cos\left(\pi +\frac{2\pi }{3}\right)$
$=-2cos\frac{2\pi }{3}=-2\left(-\frac{1}{2}\right)=1$