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The Fundamental Theorem of Algebra

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.

Proof of Theorem 6.19 is in the book.
Complex Analysis for Mathematics and Engineering

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.

Proof.

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

Exploration.

Example 1.  Find the zeros of the equation  .

Explore Solution for 1.

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

Exploration.

Example 2.  Find the zeros of the equation  .

Explore Solution 2.

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

Exploration.

Example 3.  Find the n zeros of the equation  .

Explore Solution 3.

Example 4.  Find the n zeros of the equation  .

Explore Solution 4.

Example 5.  Find the roots of the Chebyshev polynomial.

Explore Solution 5.

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

Explore Solution 6.

History of Complex Numbers

Fundamental Theorem of Algebra

The Next Module is

Uniform Convergence of Complex Functions

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