The sum of the real values of $x$ for which the middle term in the binomial expansion of ${(\frac{x^{3}}{3} + \frac{3}{x})}^{8}$ equals $5670$ is :

0

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4

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6

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8

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Solution

The correct option is A $0$
$5$ th term will be the middle term in the expansion of the given expression.
$T_{4 + 1} =^{8} C_{4} {(\frac{x^{3}}{3})}^{4} {(\frac{3}{x})}^{4} = 5670$

$\Rightarrow x^{8} \times \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2} = 5670$
$\Rightarrow x^{8} = 81$
$\Rightarrow x = \pm \sqrt{3}$
$∴$ Sum $= 0$

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