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Theory of expressions

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Question

$If α, β and γ are roots of 4 x^{3} + 8 x^{2} - x - 2 = 0, then the value of \frac{4 (α + 1) (β + 1) (γ + 1)}{α β γ} is$

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Solution

$\begin{array}{l} 4 x^{3} + 8 x^{2} - x - 2 = 4 (x - α) (x - β) (x - γ) \\ Putting x = - 1, we get \\ 3 = 4 (- 1 - α) (- 1 - β) (- 1 - γ) \\ or (α + 1) (β + 1) (γ + 1) = - \frac{3}{4} \end{array}$
$Also, α β γ = \frac{1}{2}$
$∴ \frac{4 (α + 1) (β + 1) (γ + 1)}{α β γ} = - 6$

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