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  4. If α, β, γ are the roots of the equation x3-x-1=0, then the value of Π((1+α/1-α)) is equal to -

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Q. If $\alpha, \beta, \gamma$ are the roots of the equation $x^3-x-1=0$, then the value of $\Pi\left(\frac{1+\alpha}{1-\alpha}\right)$ is equal to -

Complex Numbers and Quadratic Equations

Solution:

$x=\frac{1+\alpha}{1-\alpha} \Rightarrow \alpha=\frac{x-1}{x+1}$
$\Rightarrow\left(\frac{x-1}{x+1}\right)^3-\left(\frac{x-1}{x+1}\right)-1=0$
$\Rightarrow x^3+7 x^2-x+1=0 $
$\Rightarrow \Pi\left(\frac{1+\alpha}{1-\alpha}\right)=-1$.