Question

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

• A. $-7$
• B. $-5$
• C. $-3$
• D. $0$

Answer 1

The correct option is A

$-7$

Explanation for the correct option:

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

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

$\Rightarrow$$\alpha +\beta +\gamma =2,\alpha \beta +\beta \gamma +\gamma \alpha =3,\alpha \beta \gamma =4$

$\begin{array}{rcl}\therefore {\alpha }^2{\beta }^2+{\beta }^2{\gamma }^2+{\gamma }^2{\alpha }^2& =& {(\alpha \beta +\beta \gamma +\gamma \alpha )}^2–2\alpha \beta \gamma (\alpha +\beta +\gamma )\\ & =& 32–2\times 4\times \left(2\right)\\ & =& 9–16\\ & =& \mathbf{–}\mathbf{}\mathbf{7}\end{array}$

Hence, Option ‘A’ is Correct.