I feel that the question that I am posting is not a general research level question but probably some special form of it where the behaviors of the roots of polynomials are studied.
Here is my question:
For every $K$ being an odd number, the polynomial:
$$\sum_{i=\lceil{K/2}\rceil}^K \binom{K}{i}\theta^i\left(2(1-\theta)^{2K-i}-(1-2\theta)^{K-i}\right)=0$$
has a single root $\theta_0$ in $\theta\in(0,\,0.5)$ and when $K$ increases $\theta_0$ monotonically decreases to zero, i.e.,
$$\lim_{K\rightarrow \infty}\theta_0=0$$
Is there any way to deal with this problem to establish a proof that indeed the claim is true?
The polynomial above is the manipulation of the difference of two binomial c.d.f.s in the following way:
$$B(K/2;K,1-x)|_{x=\theta}-(1/2)B(K/2;K,1-x)|_{x=\theta/(1-\theta)}$$
I use the identity $$B(K/2;K,1-x)=1-B(K/2;K,x)=\sum_{i=\lceil{K/2}\rceil}^K \binom{K}{i}x^i(1-x)^{K-i}$$
Added (01.10): I simplified it using regularized incomplete beta function. The roots of the given polynomial is equal to the roots of
$$\small 2\theta^{\frac{1+K}{2}} F_1\left(\frac{1-K}{2},\frac{1+K}{2},\frac{3+K}{2},\theta\right)-\left(\frac{\theta}{1-\theta}\right)^{\frac{1+K}{2}} F_1\left(\frac{1-K}{2},\frac{1+K}{2},\frac{3+K}{2},\frac{\theta}{1-\theta}\right)=0$$
where $F_1$ is the hypergeometric function with respective parameters (how mathematica uses)
Thank you very much in advance.