6-6 The Fundamental Theorem of Algebra

6-6 The Fundamental Theorem of Algebra

6-6 The Fundamental Theorem of Algebra Objectives The Fundamental Theorem of Algebra Vocabulary Fundamental Theorem of Algebra If P(x) is a polynomial of degree n 1 with complex coefficients, then P(x) = 0 has at least one complex root.

Corollary Including complex roots and multiple roots, an nth degree polynomial equation has exactly n roots; the related polynomial function has exactly n zeros. For the equation x4 3x3 + 4x + 1 = 0, find the number of complex roots, the possible number of real roots, and the possible rational roots. By the corollary to the Fundamental Theorem of Algebra, x4 3x3 + 4x + 1 = 0 has four complex roots. By the Imaginary Root Theorem, the equation has either no imaginary roots,

two imaginary roots (one conjugate pair), or four imaginary roots (two conjugate pairs). So the equation has either zero real roots, two real roots, or four real roots. By the Rational Root Theorem, the possible rational roots of the equation are 1. Find the number of complex zeros of (x) = x5 + 3x4 x 3. Find all the zeros. By the corollary to the Fundamental Theorem of Algebra, there are five complex zeros. You can use synthetic division to find a rational zero. Step 1: Find a rational root from the possible roots of 1 and 3.

Use synthetic division to test each possible until you get a remainder of zero. 3 1 1 x4 3 0 3 0 0 0 0

0 0 1 3 0 3 1 0 1 So 3 is one of the roots. (continued)

Step 2: Factor the expression x4 1. x4 1 = (x2 1)(x2 + 1) = (x 1)(x + 1)(x2 + 1) So 1 and 1 are also roots. Factor x2 1. Step 3: Solve x2 + 1 = 0. x2 + 1 = 0 x2 = 1

x =i So i and i are also roots. The polynomial function (x) = x5 + 3x4 x 3 has three real zeros of x = 3, x = 1, and x = 1, and two complex zeros of x = i, and x = i. Homework Pg 343 # 1, 2, 9, 10, 17, 18

Recently Viewed Presentations

  • Geology of the Hawaiian Islands - SOEST

    Geology of the Hawaiian Islands - SOEST

    Times New Roman Tahoma Arial Wingdings MS PGothic Helvetica Times Verdana Comic Sans MS Garamond Stream Dynamic Earth The Flow of the Continents (Chapter 5) Building Mountains: Rock Deformation and Earthquakes Rock Deformation Deformation Deformation of rocks Small-Scale Folds Stress...
  • Unit 4 - Telling Where You Live - Mrs. Inabnit's ASL Classes

    Unit 4 - Telling Where You Live - Mrs. Inabnit's ASL Classes

    How do you write sentences in ASL? We write sentences in ASL by "Glossing" them. How do you gloss? WRITE IN ALL CAPITAL LETTERS. Do not sign any linking verbs. (am, is, are, was, were, be, been, being, has, does,...
  • ECE 470 Introduction - dejazzer.com

    ECE 470 Introduction - dejazzer.com

    COEN-4730 Computer ArchitectureLecture #8 (part 2)Networks-on-Chip (NoC) CristinelAbabei. Dept. of Electrical and Computer Engr., Marquette University
  • Standards of Psychiatric-mental Health Clinical Nursing Practice

    Standards of Psychiatric-mental Health Clinical Nursing Practice

    Milieu Therapy The psychiatric-mental health nurse provides, structures, and maintains a therapeutic environment in collaboration with the client and other health care providers.
  • CASUALTY EVACUATION OPERATIONS PURPOSE The purpose of this

    CASUALTY EVACUATION OPERATIONS PURPOSE The purpose of this

    Platoon leaders should designate both a primary and alternate aid and litter team within each squad. When conducting patrols, it is possible that you could receive a casualty due to either an accident or hostile action. One rapid method of...
  • Better Practices for Integrating Educational Technology into ...

    Better Practices for Integrating Educational Technology into ...

    Computer-based multi media - Skype, Moodle, blackboard, interactive learning games. Purpose of the Study. Link current research on educational technology to data I collect from my own learning/teaching environment to create a plan to better integrate educational technology into the...
  • Logistics of Effective Implementation Roger Zoorob, MD, FAAFP

    Logistics of Effective Implementation Roger Zoorob, MD, FAAFP

    Training and implementation should have continuity. For example, if you train residents and their attending does not do it, it is a problem. Give staff a voice on how things are implemented. Buy-in from everyone involved (including front desk) is...
  • Writing the APUS DBQ - Miami-Dade County Public Schools

    Writing the APUS DBQ - Miami-Dade County Public Schools

    The DBQ Like the standard essay, however, the DBQ will also be judged on its thesis, argument, and supporting evidence. contained in the DBQ are unlikely to be the familiar classics (the Emancipation Proclamation or Declaration of Independence, for example),...