Question

If $\alpha ,\beta ,\gamma$ are the roots of the equation $x^3+4x+2=0$, then ${\alpha }^3+{\beta }^3+{\gamma }^3=$ ?

• A. $2$
• B. $6$
• C. $-2$
• D. $-6$

Answer 1

The correct option is D $-6$

$\alpha ,\beta ,\gamma$ are the roots of $x^3+4x+2=0$

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

$\alpha \beta +\beta \gamma +\alpha \gamma =4$

$\alpha \beta \gamma =-2$

We know that $a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ac)$

Similarly, $(\alpha {)}^3+(\beta {)}^3+(\gamma {)}^3-3\alpha \beta \gamma =(\alpha +\beta +\gamma \left)\right[(\alpha +\beta +\gamma {)}^2-3(\alpha \beta +\beta \gamma +\alpha \gamma \left)\right]$

$\Rightarrow (\alpha {)}^3+(\beta {)}^3+(\gamma {)}^3+6=0$

$\Rightarrow (\alpha {)}^3+(\beta {)}^3+(\gamma {)}^3=-6$