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

$\displaystyle x=2+\sqrt{3}, x=1-i, x=1$. Thus, you have four factors as follows (by the complex conjugate theorem): $\displaystyle (x-1)(x-(2+\sqrt{3}))(x-(1-i))(x-(1+i))$ which more elagantly written is $\displaystyle (x-1)((x-2)-\sqrt{3})((x-1)+i)((x-1)-i).$.

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 $\displaystyle 2-\sqrt{3}, )$?

Hmm...