The sum of the real values of x for which the middle term in the binomial expansion of (x^…

Question

The sum of the real values of $x$ for which the middle term in the binomial expansion of $\left(\dfrac{x^3}3+\dfrac3x\right)^8$ equals 5670 is:

Accepted Answer

Middle term (5th, since $n=8$): $T_5=\binom84\left(\dfrac{x^3}3\right)^4\left(\dfrac3x\right)^4=5670$

$$\frac{8\times7\times6\times5}{4\times3\times2}\times x^{12-4}=5670\quad\Rightarrow\quad70x^8=5670\quad\Rightarrow\quad x^8=81$$

Sum of all real roots of $x^8-81=0$ (four real roots $\pm3^{1/2},\pm i$-pairs excluded; the two real roots $x=\pm\sqrt3$) is 0.

Solutions · 1