This section continues the discussion of complex numbers that was begun with:
Much of the material on this current page is needed to prove
a beautiful property of polynomials with real number coefficients:
any nonreal zeroes must occur in complex conjugate pairs.
(More on this in a future section!)

$z := a + bi$ $\overline{z} = a  bi$ A complex number $\,z\,$ can be represented: $\bullet$ as a point $\bullet$ as a vector from the origin to the point The length of the vector representing $\,z\,$ is called the modulus of $\,z\,$ and is denoted by $\,z\,$. Equivalently, $\,z\,$ represents the distance of the complex number from the origin in the complex plane. From the Pythagorean Theorem: $z = \sqrt{a^2 + b^2}$ 
Properties that ‘look like’ they hold for only two complex numbers
actually hold for any finite number of complex numbers.
The key is to take two things and treat them as a single thing, as shown next:
$$
\begin{alignat}{2}
\cssId{s111}{\overline{u + v + w}} \ &
\cssId{s112}{= \overline{(u + v) + w}}
&&\qquad\cssId{s113}{\text{treat $\,u+v\,$ as a singleton}}\cr\cr
&\cssId{s114}{= \overline{u+v} + \overline{w}}
&&\qquad\cssId{s115}{\text{it works for two}}\cr\cr
&\cssId{s116}{= \overline{u} + \overline{v} + \overline{w}}
&&\qquad\cssId{s117}{\text{it works for two}}
\end{alignat}
$$
Since it works for two, it works for three.
Repeating the process—since it works for three, it works for four.
And so on!
Thus:
IDEA OF PROOF:
The ideas that make this work are illustrated with a quadratic polynomial,
but it should be clear that they work for a polynomial of any degree.
Let $\,P(x) = ax^2 + bx + c\,$, where the coefficients are real numbers and $\,a\ne 0\,$.
Let $\,z\,$ be a complex number.
Then:
$$
\begin{alignat}{2}
\cssId{s137}{\overline{P(z)}} \
&\cssId{s138}{= \overline{az^2 + bz + c}} \qquad
&&\cssId{s139}{\text{definition of $P$}}\cr\cr
&\cssId{s140}{= \overline{az^2} + \overline{bz} + \overline{c}}
&&\cssId{s141}{\text{the conjugate of a sum is the sum of the conjugates}}\cr\cr
&\cssId{s142}{= \overline{a}\overline{z^2} + \overline{b}\overline{z} + \overline{c}}
&&\cssId{s143}{\text{the conjugate of a product is the product of the conjugates}}\cr\cr
&\cssId{s144}{= a\,\overline{z^2} + b\,\overline{z} + c}
&&\cssId{s145}{\text{the conjugate of a real number is itself}}\cr\cr
&\cssId{s146}{= a\,{\overline{z}}^2 + b\,\overline{z} + c}
&&\cssId{s147}{\text{the power property for conjugates}}\cr\cr
&\cssId{s148}{= P\,(\,\,\overline{z}\,)}
&&\cssId{s149}{\text{function evaluation; definition of $\,P\,$}}
\end{alignat}
$$
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
