Finding a Polynomial Function

Suppose you're given the following zeroes of a polynomial of degree 5, and leading coefficient 1:

. Thus, you have four factors as follows (by the complex conjugate theorem): which more elagantly written is .

Knowing that the polynomial is degree 5 with a leading coefficient of 1, how do we determine the remaiing zero?

I have some ideas, but I'm not exactly sure. Do we expand out the factors we have and simply make up a factor that will give you a leading co-efficient of 1 and then expand to create the 5th degree polynomial in expanded form? Do we say that there is an additional zero of ?

Hmm...