Proving the non-existence of rational zeros in a polynomial with irrational coefficients

Question

I have a conjecture, but have no idea how to prove it or where to begin. The conjecture is as follows:

A polynomial with all real irrational coefficients and no greatest common factor has no rational zeros.

This conjecture excludes the cases where the polynomial does have a greatest common factor despite having an irrational coefficient, such as $x^3+\pi x^2=0$, as that has rational zero $0$.

I know that not all polynomials with rational coefficients have rational zeros, but I am not sure how to begin. How would I go about beginning to prove this? Has it already been proved - or is there a counterexample that I am missing?

Answer 1

For the case of just one irrational coefficient $a_i$, supose by absurd that there is a rational solution $q$. Then: $$(a_i=a_0+...+a_n q^n)\frac{1}{q^i}$$ hence $a_i$ is rational. Therefore for this case there is no rational solutions.

Answer 2

If one were able to prove that, that would imply that $\pi x-e=0$ has no rational roots. However, it is not known whether $e/\pi$ is rational. So, I think your conjecture as stated would be difficult to prove.