Question

If $\alpha ,\beta$ and $\gamma$ are the roots of $x^3+bx+c=0$, then ${\alpha }^2\beta +\alpha {\beta }^2+{\beta }^2\gamma +\beta {\gamma }^2+{\gamma }^2\alpha +\gamma {\alpha }^2$

• A. $c$
• B. $-c$
• C. $-3c$
• D. $3c$

Answer 1

The correct option is D

$3c$

Explanation for the correct option:

Find the value of ${\alpha }^2\beta +\alpha {\beta }^2+{\beta }^2\gamma +\beta {\gamma }^2+{\gamma }^2\alpha +\gamma {\alpha }^2$:

Given, $\alpha ,\beta$ and $\gamma$ are the roots of $x^3+bx+c=0$

$\Rightarrow$$\alpha +\beta +\gamma =0,\sum \alpha \beta =+b,\alpha \beta \gamma =–c$

$\begin{array}{rcl}\therefore {\alpha }^2\beta +\alpha {\beta }^2+{\beta }^2\gamma +\beta {\gamma }^2+{\gamma }^2\alpha +\gamma {\alpha }^2& =& \alpha \beta (\alpha +\beta )+\beta \gamma (\beta +\gamma )+\gamma \alpha (\gamma +\alpha )\\ & =& –\alpha \beta \gamma –\alpha \beta \gamma –\alpha \beta \gamma \\ & =& –3\alpha \beta \gamma \\ & =& \mathbf{3}\mathit{c}\end{array}$

Hence, Option ‘D’ is Correct.