If $x = 3 + 3^{1 / 3} + 3^{2 / 3}$ then $x^{3} - 9 x^{2} + 18 x - 12$ =

2

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3

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0

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5

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Solution

The correct option is C $0$
$x = 3 + 3^{\frac{1}{3}} + 3^{\frac{2}{3}}$
$\Rightarrow x - 3 = 3^{\frac{1}{3}} + 3^{\frac{2}{3}}$ ...... $(1)$
$\Rightarrow {(x - 3)}^{3} = {(3^{\frac{1}{3}} + 3^{\frac{2}{3}})}^{3}$
$\Rightarrow x^{3} - 9 x (x - 3) - 27 = 3 + 3^{2} + 3 \times 3^{\frac{1}{3}} \times 3^{\frac{2}{3}} (3^{\frac{1}{3}} + 3^{\frac{2}{3}})$
$\Rightarrow x^{3} - 9 x^{2} + 27 x - 27 = 3 + 9 + 3^{1 + \frac{1}{3} + \frac{2}{3}} (x - 3)$ from $(1)$
$\Rightarrow x^{3} - 9 x^{2} + 27 x - 27 = 12 + 9 x - 27$
$\Rightarrow x^{3} - 9 x^{2} + 18 x - 12 = 0$

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