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Quantitative Aptitude

Equations

If α,β,γ ar...

Question

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

A

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

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B

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

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C

q^{3} + 3 p q r + 3 r^{2}

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D

q^{3} - 3 p q r + 3 r^{2}

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Solution

The correct option is D $q^{3} - 3 p q r + 3 r^{2}$
Given expression is $x^{3} + p x^{2} + q x + r = 0$
$α + β + γ = - p α β + β γ + γ α = q α β γ = - r {(α β + β γ + γ α)}^{3} = q^{3} {(α β + β γ)}^{3} + {(α γ)}^{3} + 3 (α β + β γ) (α γ) (α β + β γ + γ α) = q^{3} \sum α^{3} β^{3} + 3 α β^{2} γ (α β + β γ) + 3 q (- r) (α + γ) = q^{3} \sum α^{3} β^{3} - 3 r β (q - α γ) - 3 q r (- p - β) = q^{3} \sum α^{3} β^{3} + 3 p q r - 3 q r β + 3 r (α β γ) + 3 q r β = q^{3} \sum α^{3} β^{3} + 3 p q r - 3 r^{2} = q^{3} \sum α^{3} β^{3} = q^{3} - 3 p q r + 3 r^{2}$

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