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If $\alpha ,\beta ,\gamma$ are roots of $x^{3}-x+2= 0$ then the equation whose roots are $\alpha +\beta +2\gamma ,\beta +\gamma +2\alpha$ & $\gamma +\alpha +2\beta$ will be

Option 1)
$x^{3}+x-2= 0$

Option 2)
$x^{3}+x+2= 0$

Option 3)
$x^{3}-x+2= 0$

Option 4)
$x^{3}-x-2= 0$

Answers (1)

best_answer

Let $y=\alpha +\beta +2\gamma =\alpha +\beta +\gamma +\gamma =0+\gamma$

$\Rightarrow \: \gamma =y$

now, $\gamma$ will satisfy the given equation so $y^{3}-y+2=0$

$\therefore$ equation is $x^{3}-x+2=0$

$\therefore$ Option (C)

Transformation of equation -

To find equation whose roots are symmetrical functions of $\alpha$ and $\beta$ , Where $\alpha$ & $\beta$ are roots of some other equation.

- wherein

Take any of the root to be equal to $y$ & calculate $\alpha$ or $\beta$ accordingly in terms of $y$ & satisfy the given equation to get required equation.

Option 1)

$x^{3}+x-2= 0$

This is incorrect

Option 2)

$x^{3}+x+2= 0$

This is incorrect

Option 3)

$x^{3}-x+2= 0$

This is correct

Option 4)

$x^{3}-x-2= 0$

This is incorrect

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