If $α, β, γ$ are the roots of the equation $x^{3} + p x^{2} + q x + r = 0$ , then find the value of $(α - \frac{1}{β γ}) (β - \frac{1}{γ α}) (γ - \frac{1}{α β})$ .

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Solution

$x^{3} + p x^{2} + q x + r = 0$
$α β γ = \frac{- d}{a} = - r$
Now, $(α - \frac{1}{β γ}) (β - \frac{1}{γ α}) (γ - \frac{1}{α β})$
$= (\frac{α β γ - 1}{β γ}) (\frac{α β γ - 1}{γ α}) (\frac{α β γ - 1}{α β})$
$= \frac{{(α β γ - 1)}^{3}}{{(α β γ)}^{2}}$
$= \frac{{(- r - 1)}^{3}}{{(- r)}^{2}}$
$= - \frac{{(r + 1)}^{3}}{r^{2}}$

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