Find the equation whose roots are the cubes of the roots of $x^{3} + 3 x^{2} + 2 = 0$

x^{3} + 33 x^{2} + 12 x - 8 = 0

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x^{3} + 33 x^{2} + 12 x + 8 = 0

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x^{3} + 33 x^{2} - 12 x + 8 = 0

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x^{3} - 33 x^{2} + 12 x - 8 = 0

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Solution

The correct option is B $x^{3} + 33 x^{2} + 12 x + 8 = 0$
Replace $x$ by $x^{\frac{1}{3}}$ then given equation becomes $3 x^{\frac{2}{3}} = - (x + 2)$
On cubing both sides, we get $27 x^{2} = (- (x + 2))^{3}$
$\Rightarrow 27 x^{2} = - (x^{3} + 6 x^{2} + 12 x + 8)$
$\Rightarrow x^{3} + 33 x^{2} + 12 x + 8 = 0$
$∴$ The required equation is $x^{3} + 33 x^{2} + 12 x + 8 = 0$

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