If $f(x) = x^{3} + \alpha x^{2} + \beta x + \gamma$, where $\alpha,\beta,\gamma$ are rational number

Question

If $f(x) = x^{3} + \alpha x^{2} + \beta x + \gamma$, where $\alpha,\beta,\gamma$ are rational numbers and two roots of $f(x) = 0$ are eccentricities of a parabola and a rectangular hyperbola, then $\alpha + \beta + \gamma$ is equal to :

Accepted Answer

Sol. (a)$$\begin{matrix} (x - \sqrt{2})(x - 1)(x + \sqrt{2}) = \left( x^{2} - 2 ight)(x - 1) \ = x^{3} - x^{2} - 2x + 2 \ \alpha + \beta + \gamma = - 1 - 2 + 2 = - 1 \end{matrix}$$

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