If $\alpha ,\beta ,\gamma$ are the  roots of the equation $x^3+x+1=0$ then the value of $x^3+{\beta }^3+{\gamma }^3$ is

We have, $\alpha +\beta +\gamma =0$ and $\alpha \beta \gamma =-1$

Now,

$\begin{array}{l}\alpha +\beta +\gamma =0\\ \Rightarrow {\alpha }^3+{\beta }^3+{\gamma }^3=3\alpha \beta \gamma \end{array}$

$\therefore {\alpha }^3+{\beta }^3+{\gamma }^3=3\times (-1)=-3\left[\because \alpha \beta \gamma =-1\right]$