Question

If $\alpha$ and $\beta$ are the roots of the equation $x^2-2x+4=0$, then ${\alpha }^9+{\beta }^9=$

• A. $-2^8$
• B. $2^9$
• C. $-2^{10}$
• D. $2^{10}$

Answer 1

The correct option is C $-2^{10}$

$x^2-2x+4=0$
$(x-1{)}^2-1+4=0$
$(x-1{)}^2+3=0$
$(x-1{)}^2=-3$
$(x-1)=\pm \sqrt{3}i$
$x=1\pm \sqrt{3}i$
$x=2\left(\frac{1\pm \sqrt{3}i}{2}\right)$
$=-2w,-2w^2$
Hence, $\alpha =-2w$, $\beta =-2w^2$
${\alpha }^9+{\beta }^9$
$=(-2w{)}^9+(-2w^2{)}^9$
$=-2^9(w^9+w^{18})$
Now $w^{3n}=1$ where $n\in N$
Hence, $-2^9(w^9+w^{18})$
$=-2^9\left(2\right)$
$=-2^{10}$