Question

If $\alpha ,\beta$ and $\gamma$. are the roots of the ,equation $x^3-8x+8=0$, then $\sum {\alpha }^2$ and $\sum \frac{1}{\alpha \beta }$ are respectively $=$

• A. 0 and -16
• B. 16 and 8
• C. -16 and 0
• D. 16 and 0

Answer 1

Answer: (D)

Since $\alpha ,\beta$ and $\gamma$ are the roots of the equation $x^3-8x+8=0$, 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))