Algebra I / Elem. Algebra
Introduction to Algebra Linear Equations and Inequalities Functions and Graphs I Lines and thier Graphs Linear Systems
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Exponents & Polynomials Intermediate Algebra starts here!

Factoring Rational Expressions Rational Equations and Applications
Algebra III / Inter. Algebra
Radical Expressions Nonlinear Equations and Applications Functions and Graphs II Exponential and Logarithmic Functions
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Equations and Inequalitites Functions and Graphs Polynomial and Rational Functions Exponential and Logarithmic Functions Systems and Matrices Geometry Basics Conic Sections Sequences and Series
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The Six Trigonometric Functions Right Triangle Trigonometry Circular Functions Graphs of Trigonometric Functions Trigonometric Identities Trigonometric Equations Oblique Triangles and the Laws Vectors Complex, Parametric, and Polar Forms
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15.Functions of Several Variables
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S = contains supplemental resources
Course: College Algebra
Topic: Polynomial and Rational Functions
Subtopic: Polynomial Functions III - Complex Zeros, Fundamental Thm of Algebra, Descartes Rule

Overview

Completing our study of polynomial functions, in this lesson we turn to complex zeros which will enable us to factor ANY polynomial. The Fundamental Theorem of Algebra shows that the degree of the polynomial matches the number of its roots when factored in the complex realm. Descartes Rule of Signs classifies the possible roots. The Rational Roots Theorem helps us list all the possible roots and synthetic division helps us find the actual roots.

Objectives

By the end of this topic you should know and be prepared to be tested on:

• 3.4.1 Factor polynomial involving Complex roots/factors
• 3.4.2 Know that Complex roots come in conjugate pairs
• 3.4.3 Understand the Fundamental Theorem of Algebra and know that it assures every polynomial factors down to a product of linear factors
• 3.4.4 Write all the possible rational roots of a polynomial according to the Rational Roots Theorem
• 3.4.5 Classify the possible roots of a polynomial according to Descartes Rule of Signs and display the conclusions in a Descartes Root Chart
• 3.4.6 Use the list of possible rational roots and synthetic division to systematically find roots of a polynomial and factor it completely according to FTA
• 3.4.7 Use Descartes theorems and the Upper/Lower Bounds Theorem to organize and make more efficient the process described in objective 3.3.6
• 3.4.8 Electronically graph the polynomial function to verify the roots found algebraically

Terminology

Define: complex zeros, conjugate pairs, Fundamental Theorem of Algebra, Rational Roots Theorem, Descartes, Descartes Rule of Signs, Descartes Root Chart, Upper/Lower Bounds Theorem