The equation whose roots are $2$ times the roots of $x^{3} + 3 x^{2} - 5 x + 1 = 0$ , is:

x^{3} + 6 x^{2} - 20 x + 8 = 0

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x^{3} + 6 x^{2} + 20 x + 8 = 0

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8 x^{3} - 12 x^{2} - 10 x + 1 = 0

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8 x^{3} + 12 x^{2} - 10 x + 1 = 0

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Solution

The correct option is A $x^{3} + 6 x^{2} - 20 x + 8 = 0$
The given equation is
$f (x) = x^{3} + 3 x^{2} - 5 x + 1 = 0$
The required equation is
$f (\frac{x}{2}) = {(\frac{x}{2})}^{3} + 3 {(\frac{x}{2})}^{2} - 5 (\frac{x}{2}) + 1 = 0$

$= \frac{x^{3}}{8} + \frac{3 x^{2}}{4} - \frac{5 x}{2} + 1 = 0$
On solving the equation, we get
$x^{3} + 6 x^{2} - 20 x + 8 = 0$

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