We're going over some rational root theorem stuff in class. Heres some Q's

If I use the rational root theorem to find possible roots, but none of those roots exist, is the polynomial unable to be factored? So, if the polynomial has no roots it cant be factored?

Not at all: has no rational roots, and still it can be reduced (or factored). From the non-existence of roots you can deduce irreducibility ONLY when the degree of the polynomial is
Another Q-

Today we looked at factor rings of polynomials over a field. I know the theorem:

is irreducible over

iff

.

...if and only if....WHAT!??
Now, say Im looking at a field order 25. I then consider

. Heres the question. Can i just guess and check polynomials to see if they're irreducible in Z5 and put that in for

? or is there a certain method to this?