When evaluating a polynomial at a rational number, is there an identity which would allow to separately evaluate it at the numerator and denominator? More formally, is there a rule to expand $f(a/b)$ into separate evaluations of some kind at $f(a)$ and $f(b)$?
It is possible to evaluate $f(a)$ with Horner's method and $f(1/b)$ equals to $rev(f)(b)/b^{degree(f)}$ (where $rev$ reverses the order of coefficients). What I'm struggling with is how to rewrite $f(a/b)$?
For example for exponentiation, $x^{a-b}$ can be rewritten as $x^a/x^b$. Is there something similar for polynomial evaluation?