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

The correct option is D $16$ and $0$

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

$\alpha +\beta +\gamma =0,$ $\alpha \beta +\beta \gamma +\gamma \alpha =-8,$ $\alpha \beta \gamma =-8$ ....(i)

Therefore, ${(\alpha +\beta +\gamma )}^2=0$

$\Rightarrow {\alpha }^2+{\beta }^2+{\gamma }^2+2(\alpha \beta +\beta \gamma +\gamma \alpha )=0$

$\Rightarrow \sum {\alpha }^2=-2(-8)$ .....[from equation (i)]

$=16$

And $\frac{1}{\alpha \beta }+\frac{1}{\beta \gamma }+\frac{1}{\gamma \alpha }=\frac{\gamma +\alpha +\beta }{\alpha \beta \gamma }$

$\Rightarrow \frac{1}{\sum \alpha \beta }=\frac{0}{-8}=0$ ......[from equation (i)]