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Arithmetic Mean

If α, β, γ ...

Question

If $α, β, γ$ are the roots of the equation $x^{3} + p x^{2} + q x + r = 0$ then the coefficient of $x$ in the cubic equation whose roots are $α (β + γ), β (γ + α)$ and $γ (α + β)$ is

2 q

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q^{2} + p r

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p^{2} - q r

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r (p q - r)

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Solution

The correct option is D $q^{2} + p r$
The general equation from roots is
$(x - x_{1}) (x - x_{2}) (x - x_{3}) = 0$
$= x^{3} - (α + β + γ) x^{2} + (α β + β γ + γ α) x - α β γ = 0$

therefore $p = - (α + β + γ)$ and $q = α β + β γ + γ α$

$r = - α β γ$

therefore the second equation is

$x^{3} - x^{2} (α (β + γ) + β (γ + α) + γ (α + β)) + x (α β (β + γ) (γ + α) + β γ (γ + α) (α + β) + α γ (α + β) (β + γ) - α β γ (α + β) (β + γ) (γ + α) = 0$

therefore arranging the coefficient of x
$α^{2} β^{2} + β^{2} γ^{2} + α^{2} γ^{2} + 3 α β γ (α + β + γ)$ .....................(i)

$q^{2} = α^{2} β^{2} + β^{2} γ^{2} + γ^{2} α^{2} + 2 α β γ (α + β + γ)$

the expression (i) can be written as

$q^{2} + α β γ (α + β + γ)$

$= q^{2} + p r$

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