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  4. If α, β , γ are the roots of the equation x3 - 3x2 + 2x - 1 = 0 then the value of [(1 - α) (1 -β )(1 - γ)] is

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Q. If $\alpha, \beta , \gamma$ are the roots of the equation $x^3 - 3x^2 + 2x - 1 = 0$ then the value of $[(1 - \alpha) (1 -\beta )(1 - \gamma)]$ is

COMEDKCOMEDK 2012Complex Numbers and Quadratic Equations

Solution:

We have; $\alpha, \beta , \gamma = 3$,
$\alpha \beta + \beta \gamma +\gamma \alpha = 2$ and $\alpha \beta \gamma = 1 $
Now, $( 1 - \alpha)(1 - \beta )(1- \gamma)$
$ = (1 - \beta - \alpha + \alpha \beta ) (1 - \gamma)$
$ = (1 - \beta - \alpha + \alpha \beta - \gamma + \beta \gamma + \alpha \gamma - \alpha \beta \gamma)$
$ = [1 - ( \alpha + \beta + \gamma ) + ( \alpha \beta + \beta \gamma + \gamma \alpha) - \alpha \beta \gamma]$
$ = [ 1 - 3 + 2 - 1] = - 1$