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

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Aicharya · Accepted Answer

Let $\alpha, \beta, \gamma$ are roots of equation $ \begin{aligned} & x^3+a x^2+b x+c=0 \ & \alpha+\beta+\gamma=-a \ & 	herefore \quad \alpha \beta+\beta \gamma+\gamma \alpha=b \ & 	ext { 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}=-\frac{b}{c} \end{aligned} $

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+a x^2+b x+c=0$, then…

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