Question:
Let $\alpha$ and $\beta$ be two roots of the equation $x^{2}+2 x+2=0$, then $\alpha^{15}+\beta^{15}$ is equal to :
Correct Option: , 3
Solution:
We have
$(x+1)^{2}+1=0$
$\Rightarrow(x+1)^{2}-(i)^{2}=0$
$\Rightarrow(x+1+i)(x+1-i)=0$
$\therefore \quad \mathrm{x}=\underset{\substack{a(k t)}}{-(1+\mathrm{i})}-\underset{\substack{\text { p(let) }}}{ }(1-\mathrm{i})$
So, $\quad \alpha^{15}+\beta^{15}=\left(\alpha^{2}\right)^{7} \alpha+\left(\beta^{2}\right)^{7} \beta$
$=-128(-i+1+i+1)$
$=-256$
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