if you have a polynomial of odd degree and the derivative is always increasing, then there is at most one real root. Then after you see one rational root you dont need to continue to check the others.
Otherwise: if you find a rational root try to factor it out of the original polynomial. for instance if 1 is a root then factor out (x-1). Perhaps this leaves you with a quadratic or something where you can see the roots quickly.
In general: i dont think so :O(