I am having trouble with finding the equation of a polynomial function, particularly because I do not know what any of the solutions (zeros) are.

However, I do know the following things:

1) The degree of the function

2) Coordinates for all local extrema and points of inflection

3) The end-behavior (in this case it is a positive even function, where both ends point up)

4) The Y-intercept (there is no X-intercept as there are no real solutions)

5) A number of other points

Note: There are no real solutions (zeros). I said earlier that I do not know what any of the zeros are (and I don't), I just know that there are no real ones.

I don't need an exact answer or an exact process, I just need to know how to approach this problem.

The frustrating thing is that every website I've seen either says a) how to find zeros given an equation, or b) how to find an equation given the zeros.