Here's the problem statement:
Consider all polynomials of a complex variable, $P(z)=4z^4+az^3+bz^2+cz+d$, where $a,b,c,$ and $d$ are integers, $0\le d\le c\le b\le a\le 4$, and the polynomial has a zero $z_0$ with $|z_0|=1.$ What is the sum of all values $P(1)$ over all the polynomials with these properties? (2012 AMC 12B #23)
In the solution, they state:
$z_0 = 1$ is not a possible root. $z_0 = -1$ contributes 60 to the desired sum.
Next, if $P(z)$ has no real root $z_0$ with $|z_0| = 1$, then $P(z)$ must have a non-real root with $|z_0| = 1$. Because $P(z)$ has all real coefficients, it follows that the conjugate of this $z_0$ must be a root as well. Thus, $P(x)$ has a factor of the form $z^2 - 2ez + 1$, where $e$ is a real number with $|e| <= 1$.
I'm not quite getting the part about the reasoning behind the 'factor of the form' $z^2 - 2ez + 1$. Could someone please explain, or offer an alternative solution?