Attempting to apply Rational Root Theorem - What did I do wrong?

Question

I was studying This question about the roots of the Polynomial: $$p(x)=x^{20}+(1-x)^{20}-20$$ I tried to apply the Rational Root Theorem on the expanded version: $$p(x)=2x+...-19$$.

Based on my understanding of the theorem, a root will exist among $\pm ((19/2), (2/19),2,19)$ However, these are not roots, and Wolfram Alpha says that one of the roots is: $1.161586$ (is correct to 3 digits behind the decimal point only).

I have 2 issues now:

A - Values in the list suggested by the theorem does not include any root.

B - I expect that this value ,$1.161586$, found by Wolfram Alpha to belong to the list of roots the theory suggests, but this is not the case.

I assume that rounding has nothing to do with these issues.

Did I apply the theory wrong? if so, what is the correct way to use it? Thanks.

Answer 1

The rational root theorem for a polynomial $ax^n+\cdots+b$ states that rational numbers of the form $\pm q/p$ where $q\mid b$ and $p\mid a$ could be roots of the polynomial $ax^n+\cdots+b$, and that if the polynomial has rational roots, they will be in this list. So, in this case, only $\pm 19$ and $\pm 19/2$ would be possible roots. You can check and note that none of them work, and the conclusion would then be that there are no rational numbers that satisfy the given polynomial. However, this implies that if there are roots, they will be irrational.