"find the fourth-degree polynomial function with zeros 1, -1, i and -i."

The associated factors of this poly are (x-1), (x+1), (x - i) and (x+i).

Multiply these out.  Note that (x-1)(x+1) = x^2 - 1, and that
(x-i)(x+i) = x^2 + 1.  So, mult. (x^2 -1) and (x^2 + 1) together.  Arrange the resulting poly so that powers of x are in descending order, beginning with x^4.