Question

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

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

Answer 1

Answer: (A)

$-7$

As $\alpha ,\beta ,\gamma$ are roots of $x^3-2x^2+3x-4=0$ gives
$\alpha +\beta +\gamma =2\alpha \beta +\beta \gamma +\gamma \alpha =3$
Therefore,

${\alpha }^2{\beta }^2+{\beta }^2{\gamma }^2+{\gamma }^2{\alpha }^2={(\alpha \beta +\beta \gamma +\gamma \alpha )}^2-2(\alpha +\beta +\gamma )\left(\alpha \beta \gamma \right)={\left(3\right)}^2-2\left(2\right)\left(4\right)=-7$