**for**

**6.7 The Fundamental Theorem of
Algebra**

This section is a supplement to the textbook.

In Section
6.6 we developed the background (Theorems
6.13 - 6.18) for the proof of the __Fundamental
Theorem of Algebra__.

**Theorem 6.13 (**__Morera's
Theorem__**).** Let
f(z) be a continuous function in a
simply connected domain D. If for
every closed contour in D, then
f(z) is analytic in D.

**Theorem 6.14 (Gauss's Mean Value
Theorem).** If f(z)
is analytic in a simply connected domain D that
contains the circle , then

.

**Theorem 6.15 (Maximum Modulus
Principle).** Let f(z)
be analytic and nonconstant in the bounded domain D. Then does
not attain a maximum value at any point
in D.

**Theorem 6.16 (**__Maximum
Modulus
Principle__**).** Let
f(z) be analytic and nonconstant in
the bounded domain D. If f(z)
is continuous on the closed region R
that consists of D and all of its
boundary points B, then
assumes its maximum value, and does so only at point(s)
on the boundary B.

**Theorem 6.17 (Cauchy's
Inequalities).** Let f(z)
be analytic in the simply connected domain D
that contains the circle . If holds
for all points , then

for .

**Theorem 6.18 (**__Liouville's
Theorem__**).** If
f(z) is an entire function and is
bounded for all values of z in the
complex plane, then f(z) is
constant.

**Theorem 6.19 (**__Fundamental
Theorem of
Algebra__**).** If
P(z) is a polynomial of degree
,
then P(z) has at least one zero.

**Proof of Theorem 6.19 is in the book.
**

**Corollary 6.4.** Let
P(z) be a polynomial of degree
. Then
P(z) can be expressed as the product
of linear factors. That is,

where are
the zeros of P(z) counted according
to multiplicity an A is a
constant.

In Section
1.1, we introduced the formulas of __Cardano__
and __Tartaglia__.
Historically, formulas have been developed for the __quadratic
equation__, __cubic
equation__ and __quartic
equation__. There is no general formula for polynomial
equations higher than fourth degree (see __Abel's
Impossibility Theorem__).

**The solution of the cubic
equations.** The depressed cubic
equation
has roots

**Example 1.** Find the
zeros of the equation .

**The solution of the cubic
equations.** The general cubic
equation
has roots

**Example 2.** Find the
zeros of the equation .

**The solution of the quartic
equations.** *Mathematica* can construct the
solutions to the general quartic equation.

**Example 3.** Find the
n zeros of the
equation .

**Example 4.** Find the
n zeros of the
equation .

**Example 5.** Find the
roots of the
Chebyshev polynomial.

**Extra Example
6.** Find the n
zeros of the equation .

**Exercises
for Section 6.7 The Fundamental Theorem of
Algebra**** **

__Fundamental
Theorem of Algebra__

**The Next Module
is**

**Uniform
Convergence of Complex Functions**

**Return to the Complex
Analysis Modules**

__Return
to the Complex Analysis Project__

This material is coordinated with our book Complex Analysis for Mathematics and Engineering.

(c) 2012 John H. Mathews, Russell W. Howell