Number of solutions to infinite sum of polynomial equation How many solutions (at most) are there in the following equation?

$a_0+a_1x+a_2x^2+a_3x^3+\cdots=0$, where each $a_i\in\mathbb{R}$, and $x\in (0,1)$?

Is it possible to find some $\underline{x}\in (0,1) $, such that all $x\geq \underline{x}$ are the solutions to the above equation?

I know that the fundamental theorem of algebra tells us that there are $n$ solutions for the $n$ degree polynomial, but don't know that happens when $n$ is infinity.
The theorem only says there can be ATMOST n solutions for n degree polynomial. There could be much less. As your equation ends with $=0$, the no. of solutions can be given by the number of  indivisible factors of your polynomial; if there are no factors, there is 1 solution (not necessary a real no.).
The theorem only says there can be ATMOST n solutions for n degree polynomial. There could be much less. As your equation ends with $=0$, the no. of solutions can be given by the number of  indivisible factors of your polynomial; if there are no factors, there is 1 solution (not necessary a real no.).