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Transformation of Roots: Linear Combination of Roots

If α, β, γ ar...

Question

If $α, β, γ$ are the roots of $x^{3} - 3 x + 7 = 0$ , then the equation whose roots are $α + β - γ, β + γ - α, γ + α - β$ is

A

x^{3} + 12 x - 56 = 0

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B

x^{3} - 12 x + 56 = 0

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C

x^{3} - 12 x - 56 = 0

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D

x^{3} + 12 x + 56 = 0

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Solution

The correct option is C $x^{3} - 12 x - 56 = 0$
$x^{3} - 3 x + 7 = 0 \to roots are α, β, γ$
$\Rightarrow α + β + γ = 0 \Rightarrow α + β - γ = - 2 γ, β + γ - α = - 2 α, γ + α - β = - 2 β$

$∴$ The required equation whose roots are $- 2 α, - 2 β, - 2 γ$ is
$f (- \frac{x}{2}) = 0 \Rightarrow {(- \frac{x}{2})}^{3} - 3 (- \frac{x}{2}) + 7 = 0 \Rightarrow x^{3} - 12 x - 56 = 0$

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