Section 6.5 Factoring over and
Subsection 6.5.1 Multiplicative Group of Complex Numbers
The complex numbers are defined asExample 6.30.
Let
and
Also,
Example 6.33.
Suppose that
and
Hence, the rectangular representation is
Conversely, if we are given a rectangular representation of a complex number, it is often useful to know the number's polar representation. If
and
so
Proposition 6.34.
Let
Example 6.35.
If
Theorem 6.36. DeMoivre.
Let
for
Proof.
We will use induction on
Example 6.37.
Suppose that
The Circle Group and the Roots of Unity.
The multiplicative group of the complex numbers,Proposition 6.38.
The circle group is a subgroup of
Theorem 6.39.
If
where
Proof.
By DeMoivre's Theorem,
The
Example 6.40.
The 8th roots of unity can be represented as eight equally spaced points on the unit circle (Figure 6.41). The primitive 8th roots of unity are
Subsection 6.5.2 Factoring With and Without Complex Numbers
Every odd degree polynomial has a root inExample 6.42.
Factor
Example 6.43.
Factor
Example 6.44.
Factor
Example 6.45.
Factor
Exercises 6.5.3 Practice Problems
1.
Evaluate each of the following.
(a) \(-3 + 3i\text{;}\) (c) \(43- 18i\text{;}\) (e) \(i\)
2.
Convert the following complex numbers to the form
(a) \(\sqrt{3} + i\text{;}\) (c) \(-3\text{.}\)
3.
Change the following complex numbers to polar representation.
(a) \(\sqrt{2} \cis( 7 \pi /4)\text{;}\) (c) \(2 \sqrt{2} \cis( \pi /4)\text{;}\) (e) \(3 \cis(3 \pi/2)\text{.}\)
4.
Calculate each of the following expressions.
(a) \((1 - i)/2\text{;}\) (c) \(16(i - \sqrt{3}\, )\text{;}\) (e) \(-1/4\text{.}\)
5.
Prove that the function
6.
Let
7.
Is
8.
Factor
Exercises 6.5.4 Collected Homework
1.
Factor the polynomial
Do the factoring in that order.
2.
True or false: