Question

If $\alpha ,\beta ,\gamma$ are the roots of $x^3+p{x}^2+qx+r=0,$ then find the value of
$(\alpha -\frac{1}{\beta \gamma })(\beta –\frac{1}{\gamma \alpha })(\gamma –\frac{1}{\alpha \beta })$

• A. $\frac{(r^3-1)}{r^2}$
• B. $\frac{(r^3+1)}{r^2}$
• C. $\frac{-(r+1{)}^3}{r^2}$
• D. None of these

Answer 1

The correct option is C

$\frac{-(r+1{)}^3}{r^2}$

Given $\alpha ,\beta ,\gamma$ are the roots of the cubic polynomial $x^3+p{x}^2+qx+r=0$

Using the relation between the roots and coefficients, we get;
$\alpha \beta \gamma =-r$

$\Rightarrow (\alpha -\frac{1}{\beta \gamma })(\beta –\frac{1}{\gamma \alpha })(\gamma –\frac{1}{\alpha \beta })=(\frac{\alpha \beta \gamma -1}{\beta \gamma })(\frac{\alpha \beta \gamma -1}{\alpha \gamma })(\frac{\alpha \beta \gamma -1}{\beta \alpha })=\frac{(\alpha \beta \gamma -1{)}^3}{(\alpha \beta \gamma {)}^2}=\frac{-(r+1{)}^3}{r^2}$