Or maybe it assumes rational roots.

Which would be useless in context because even for quadratic polynomial problems in middle school, very often the solutions are irrational.

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.