Question

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

• A. $3l$
• B. $-3l$
• C. $-3m$
• D. $0$

Answer 1

The correct option is C $-3m$

$\alpha +\beta +\gamma =0$,
$\alpha \beta +\beta \gamma +\gamma \alpha =l$,
$\alpha \beta \gamma =-m$
and we know that
${\alpha }^3+{\beta }^3+{\gamma }^3-3\alpha \beta \gamma =(\alpha +\beta +\gamma )({\alpha }^2+{\beta }^2+{\gamma }^2-\alpha \beta -\beta \gamma -\alpha \gamma )$
$\because \alpha +\beta +\gamma =0$
$\therefore {\alpha }^3+{\beta }^3+{\gamma }^3=3\alpha \beta \gamma =-3m$