The sum of the real values of x for which the middle term in the binomial expansion of `((x^(3))/(3)+(3)/(x))^(8)` equals 5670 is

Options

A. 4
B. 0
C. 6
D. 8

Explanation

In the expansion of `((x^(3))/(3) + (3)/(x))^(8)`, the ratio term is `T_(4 + 1)`
[`:'` Here n = 8 which is even, therefore middle term `= ((n + 2)/(2))`th term]
`:. 5670 = .^(8)C_(0) ((x^(3))/(3))^(4) ((3)/(x))^(4) = (8 . 7 . 6 . 5)/(1 .2 . 3 . 4) x^(8)`
`[:' T_(r + 1) = .^(8)C_(r ) ((x^(3))/(3))^(8-r) ((3)/(x))^(t )]`
`implies x^(8) = 3^(4) implies x = +- sqrt(3)`
So, sum of all value of x i.e. `+ sqrt(3)` and - `sqrt(3) = 0`