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?