Mathematics - Binomial Theorem Question with Solution | TestHub

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, $T_{4+1}={}^8{C}_4{\left(\frac{x^3}{3}\right)}^4{\left(\frac{3}{x}\right)}^4=5670$ $\Rightarrow \frac{8\times 7\times 6\times 5}{4\times 3\times 2}\times x^{12-4}=5670$ $\Rightarrow x^8=81$ $\Rightarrow x^8-81=0$ $\therefore$ sum of all values of $x=$ sum of roots of equation $(x^8-81=$ 0).