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Common Roots

If α, β, ...

Question

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

A

2 (p^{2} - 3 q)

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B

r - p q

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C

q p + r

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D

\frac{p^{2} - 2 q}{r^{2}}

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Solution

The correct option is C $q p + r$
$x^{3} - p x^{2} + q x + r = 0$
Roots are $α, β, γ$
$α + β + γ = p$
$α β + β γ + γ α = q$
$α β γ = - r$
$(α + β) (β + α) (γ + α) = (α β + β^{2} + α γ + β γ) (γ + α)$
$α β γ + β^{2} γ + α γ^{2} + {β γ}^{2} + α^{2} β + β^{2} α + α^{2} γ + α β γ$
$= (α^{2} β + α^{2} γ + α β γ) + (β^{2} γ + β^{2} α + α β γ) + (γ^{2} α + γ^{2} β + α β γ) - α β γ$
$= α (α β + β γ + γ α) + β (α β + β γ + γ α) + γ (α β + β γ + γ α) - α β γ$
$= (α + β + γ) (α β + β γ + γ α) - α β γ$
$= p q + r$

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