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Q.
If $\alpha, \beta$ and $\gamma$. are the roots of the ,equation $x^{3}-8 x+8=0$, then $\sum \alpha^{2}$ and $\sum \frac{1}{\alpha \beta}$ are respectively $=$
KCETKCET 2006Complex Numbers and Quadratic Equations
Solution:
Since $\alpha, \beta$ and $\gamma$ are the roots of the equation $x^{3}-8 x+8=0$, then
$\alpha+\beta+\gamma=0, \alpha \beta+\beta \gamma+\gamma \alpha=-8, \alpha \beta \gamma=-8 \,\,\,\dots(i)$
$\therefore (\alpha+\beta+\gamma)^{2}=0 $
$\Rightarrow \alpha^{2}+\beta^{2}+\gamma^{2}+2(\alpha \beta+\beta \gamma+\gamma \alpha)=0 $
$\Rightarrow \Sigma \alpha^{2}=-2(-8)$ (From (i)
and $\frac{1}{\alpha \beta}+\frac{1}{\beta \gamma}+\frac{1}{\gamma \alpha}=\frac{\gamma+\alpha+\beta}{\alpha \beta \gamma}$
$\Rightarrow \frac{1}{\Sigma \alpha \beta}=\frac{0}{-8}=0$
(From (i))