Question

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

• 0

• 6

• 4

• 8

Answer 1

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 0.

Explanation:

Middle Term, `(n/2 + 1)^("th")` term in the binomial expansion of `(x^3/3 + 3/x)^8` is 

T_{4 + 1} = `""^8C_4(x^3/3)^4 (3/x)^4` = 5670

⇒ `(8 xx 7 xx 6 xx 5)/(4 xx 3 xx 2) xx x^(12  -  4)` = 5670

⇒  x⁸ = 81

⇒ x⁸ – 81 = 0

∴ Sum of all values of x = sum of roots of equation (x⁸ – 81 = 0) = 0.