Question

Let $\alpha$ and $\beta$ be two roots of the equation $x^2+2x+2=0$, then ${\alpha }^{15}+{\beta }^{15}$ is equal to

• A. $512$
• B. $-512$
• C. $-256$
• D. $256$

Answer 1

The correct option is C $-256$

$\alpha ,\beta$ are roots of equation $x^2+2x+2=0$

root$=\frac{-2\pm \sqrt{4-8}}{2}=-1\pm i$

$\Rightarrow \alpha =-1+i$ and $\beta =-(1+i)$

$\Rightarrow {\alpha }^2=-2i$ and ${\beta }^2=2i$

$\Rightarrow {\alpha }^{14}=(-2i{)}^7$ and ${\beta }^{14}=(2i{)}^7$

$\Rightarrow {\alpha }^{14}=-128i^3$ and ${\beta }^{14}=128i^3$

$=128i$ and $=-128i$

${\alpha }^{15}=128i(i-1)$ and ${\beta }^{15}=128i(i+1)=-128+128i$

$=-128-128i$

$\Rightarrow {\alpha }^{15}+{\beta }^{15}=-128-128i-128+128i=-256$.