(a) We have, ${x^3} + px + q = 0$ …..$(i)$

$\therefore $ The roots of equation $(i)$ is $\alpha ,\,\beta $ and $\gamma $

$\therefore $ The sum of roots = $\alpha + \beta + \gamma $

= $\frac{{{\rm{Coefficient of }}{x^2}}}{{{\rm{Coefficient of }}{x^3}}} = \frac{{ – 0}}{1} = 0$

and the product of any two roots

= $\alpha \beta + \beta \gamma + \gamma \alpha = \frac{{{\rm{Coefficient of }}x}}{{{\rm{Coefficient of }}{x^3}}} = p$

$\therefore $ Product of all three roots = $\alpha \beta \gamma $= ${t_1} = (d/{v_1})$

$\alpha + \beta + \gamma = 0$

$\therefore $ ${\alpha ^3} + {\beta ^3} + {\gamma ^3} = 3\alpha \beta \gamma = – 3q$.