The sum of the real values of $x$ for which the middle term in the binomial expansion of…
The sum of the real values of $x$ for which the middle term in the binomial expansion of $\left(\frac{x^{3}}{3}+\frac{3}{x}\right)^{8}$ equals 5670 is :
0
6
4
8
Solution
Middle Term, $\left(\frac{n}{2}+1\right)^{\text {th }}$ term in the binomial
expansion of $\left(\frac{x^{3}}{3}+\frac{3}{x}\right)^{8}$ is,
$T_{4+1}={ }^{8} C_{4}\left(\frac{x^{3}}{3}\right)^{4}\left(\frac{3}{x}\right)^{4}=5670$
$\Rightarrow \quad \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2} \times x^{12-4}=5670$
$\Rightarrow \quad x^{8}=81$
$\Rightarrow x^{8}-81=0$
$\therefore$ sum of all values of $x=$ sum of roots of equation $\left(x^{8}-81=\right.$ 0).