In both cases the rational roots theorem works.

The above polynomial will only have rational roots if those rational roots are of the form:

To be precise the numerator is factors of the constant term, the denominator is factors of the coefficient of the leading term.

So in this case we only have four possible rational roots: . Check these and you have your answer.

The same works for the polynomial mod(2):

(mod 2)

Here we only have two possibles: .

-Dan