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Greatest Integer Function

If α,β,γ ar...

Question

If $α, β, γ$ are the roots of $x^{3} + p x^{2} + q x + r = 0$ , then $(β + r - 3 α) (γ + α - 3 β) (α + β - 3 γ) =$

A

3 p^{3} + 16 p q + 64 r

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B

3 p^{3} - 16 p q + 64 r

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C

3 p^{3} - 16 p q

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D

3 p^{3} + 16 p q

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Solution

The correct option is A $3 p^{3} - 16 p q + 64 r$
As $α, β, γ$ are the roots of $x^{3} + p x^{2} + q x + r = 0$
Gives
$s_{1} = α + β + γ = - p s_{2} = α β + β γ + γ α = q s_{3} = α β γ = - r$
Now $α + β + γ = - p \Rightarrow α + β = - p - γ \Rightarrow β + γ = - p - α \Rightarrow α + γ = - p - β$

Using this we get
$(β + γ - 3 α) (γ + α - 3 β) (α + β - 3 γ) = - (p + 4 α) (p + 4 β) (p + 4 γ) = - (p^{3} + 4 p^{2} (α + β + γ)) + 16 p (α β + β γ + γ α) + 64 (α β γ) = - (p^{3} + 4 p^{2} (- p)) + 16 p (q) + 64 (- r) = 3 p^{3} - 16 p q + 64 r$

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