In which case it could say "no rational solutions found"

Yeah, I could also write a program that checks if all solutions are 0 and output the factorization, otherwise say "there are non-zero solutions". That's far from "a program that factors polynomials".

If you brag about how "you can design a computer program to factor any polynomial equation string input to it" in a class about quadratic equations and your code can't factor x^2 - 2, that's just not very impressive, regardless of your age.