Sunday, 4 October 2015

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

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?

No comments:

Post a Comment

real analysis - How to find $lim_{hrightarrow 0}frac{sin(ha)}{h}$

How to find $\lim_{h\rightarrow 0}\frac{\sin(ha)}{h}$ without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...