[14 points] We want to find a nonzero polynomial $p ( x )$ with integer coefficients having the following property.
$$\text { Letting } q ( x ) : = \frac { p ( x ) } { x ( 1 - x ) } , \quad q ( x ) = q \left( \frac { 1 } { 1 - x } \right) \text { for all } x \notin \{ 0,1 \}$$
(i) Find one such polynomial with the smallest possible degree.\\
(ii) Find one such polynomial with the largest possible degree OR show that the degree of such polynomials is unbounded.