ordering two rational functions

Question

I have read that $$\frac{x^2 +3}{2x+1}$$ is less than $$\frac{2x-1}{2x+1}$$ in an ordered field,in $\mathbb{Q}((x))$, but how is that result computed? How do we compare two rational fractions like this? Is it to do with the highest exponents? Any thoughts would help greatly, thank you...

Answer 1

There is an ordering of rational functions given by $$\frac{p(x)}{q(x)}>0\Leftrightarrow\frac{a_0}{b_0}>0$$ where $a_0,b_0$ are the leading coefficients of $p(x),q(x)$ respectively.