Question

If $\alpha ,\beta ,\gamma$ are roots of the equation $x^3+a{x}^2+bx+c=0,$ then${\alpha }^{-1}+{\beta }^{-1}+{\gamma }^{-1}=$

• A. $\frac{c}{a}$
• B. $\frac{a}{c}$
• C. $\frac{b}{a}$
• D. $-\frac{b}{c}$

Answer 1

The correct option is D

$-\frac{b}{c}$

$\alpha ,\beta ,\gamma$ are the roots of the equation

$x^3+a{x}^2+bx+c=0$ so $\alpha +\beta +\gamma =-a,\alpha \beta +\beta \gamma +\gamma \alpha =b$ and $\alpha \beta \gamma =-c$
now ${\alpha }^{-1}+{\beta }^{-1}+{\gamma }^{-1}=\frac{1}{\alpha }+\frac{1}{\beta }+\frac{1}{\gamma }=\frac{\alpha \beta +\beta \gamma +\gamma \alpha }{\alpha \beta \gamma }=-\frac{b}{c}$