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Q.
If $\alpha, \beta, \gamma$ are the roots of $x^{3}-2 x^{2}+3 x-4=0$ then the value of $\alpha^{2} \beta^{2}+\beta^{2} \gamma^{2}+\gamma^{2} \alpha^{2}$ is
EAMCETEAMCET 2007
Solution:
If $\alpha, \beta, \gamma$ are the roots of the equation
$x^{3}-2 x^{2}+3 x-4=0$, then
$\alpha+\beta+\gamma=\frac{2}{1}=2$
$\alpha \beta+\beta \gamma+\gamma \alpha=3$
and $\alpha \beta \gamma=4$
We know that
$\alpha^{2} \beta^{2}+ \beta^{2} \gamma^{2}+\gamma^{2} \alpha^{2}$
$=(\alpha \beta+\beta \gamma+\gamma \alpha)^{2}-2(\alpha \beta \gamma)(\alpha+\beta+\gamma)$
$=(3)^{2}-2(4)(2)$
$=9-16$
$=-7$