Zeros+of+Polynomial+Functions

media type="custom" key="3957205" Created by Laurence Kelley, Kevin Weigand, and Samantha Kraus

media type="custom" key="3962839" Sections 2.5, 2.6, and 2.7 focus on diverse properties of algebraic functions. Section 2.5 elaborates on the ways in which one solves for the zeros of polynomial functions. Section 2.6 describes the properties of rational functions, focusing specifically on different types of asymptotes. Section 2.7 portrays nonlinear inequalities, giving an extensive overview on polynomial inequalities. Objectives · Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions. · Find rational zeros of polynomial functions. · Find conjugate pairs of complex zeros. · Find zeros of polynomials by factoring. · Use Descartes’ Rule of Signs and the Upper and Lower Bound Rules to find zeros of polynomials. · Find the domains of rational functions. · Find the horizontal and vertical asymptotes of graphs of rational functions. <span style="font-size: 12pt; color: #5f497a; font-family: Symbol; mso-fareast-font-family: Symbol; msofareastfontfamily: Symbol; msolist: Ignore; msothemecolor: accent4; msothemeshade: 191; mso-themecolor: accent4; mso-themeshade: 191; mso-bidi-font-family: Symbol; msobidifontfamily: Symbol;">· <span style="font-size: 12pt; color: #5f497a; font-family: 'Constantia','serif'; mso-themecolor: accent4; mso-themeshade: 191;">Analyze and sketch graphs of rational functions. <span style="font-size: 12pt; color: #5f497a; font-family: Symbol; mso-fareast-font-family: Symbol; msofareastfontfamily: Symbol; msolist: Ignore; msothemecolor: accent4; msothemeshade: 191; mso-themecolor: accent4; mso-themeshade: 191; mso-bidi-font-family: Symbol; msobidifontfamily: Symbol;">· <span style="font-size: 12pt; color: #5f497a; font-family: 'Constantia','serif'; mso-themecolor: accent4; mso-themeshade: 191;">Sketch graphs of rational functions that have slant asymptotes. <span style="font-size: 12pt; color: #5f497a; font-family: Symbol; mso-fareast-font-family: Symbol; msofareastfontfamily: Symbol; msolist: Ignore; msothemecolor: accent4; msothemeshade: 191; mso-themecolor: accent4; mso-themeshade: 191; mso-bidi-font-family: Symbol; msobidifontfamily: Symbol;">· <span style="font-size: 12pt; color: #5f497a; font-family: 'Constantia','serif'; mso-themecolor: accent4; mso-themeshade: 191;">Solve polynomial inequalities. <span style="font-size: 12pt; color: #5f497a; font-family: Symbol; mso-fareast-font-family: Symbol; msofareastfontfamily: Symbol; msolist: Ignore; msothemecolor: accent4; msothemeshade: 191; mso-themecolor: accent4; mso-themeshade: 191; mso-bidi-font-family: Symbol; msobidifontfamily: Symbol;">· <span style="font-size: 12pt; color: #5f497a; font-family: 'Constantia','serif'; mso-themecolor: accent4; mso-themeshade: 191;">Solve rational inequalities. <span style="font-size: 14pt; color: #632423; line-height: 200%; font-family: 'Constantia','serif'; mso-themecolor: accent2; mso-themeshade: 128;">Table of Contents
 * **<span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Title ** || <span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi; mso-bidi-font-weight: bold;">Kevin ||
 * **<span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Voki ** || <span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Kevin ||
 * **<span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Summary ** || <span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Kevin ||
 * **<span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Picture ** || <span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Larry ||
 * **<span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Objectives ** || <span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Larry ||
 * **<span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Tables of Contents ** || <span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Sam ||
 * **<span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Title ** || <span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Sam ||
 * **<span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">“Ain’t Nothin but an E-quation” ** || <span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Larry & Kevin ||
 * **<span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Summary ** || <span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Larry ||
 * **<span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Learn Concepts with the Help of Pictures ** || <span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Sam ||
 * **<span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Picture ** || <span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Kevin ||
 * **<span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Interactive Website ** || <span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Kevin ||
 * **<span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">PowerPoint ** || <span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Sam ||
 * **<span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Glossary ** || <span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Larry ||
 * **<span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Title ** || <span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Sam ||
 * **<span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Summary ** || <span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Larry ||
 * **<span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Pictures ** || <span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Larry ||
 * **<span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Interactive Website ** || <span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Sam ||
 * **<span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">PowerPoint ** || <span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Sam ||
 * **<span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Picture ** || <span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Larry ||
 * **<span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Title ** || <span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Sam ||
 * **<span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Summary ** || <span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Kevin ||
 * **<span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">PowerPoint ** || <span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Sam ||
 * **<span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Quiz ** || <span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Larry ||
 * **<span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Kdub’s Korner ** || <span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Kevin ||
 * **<span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Podcast ** || <span style="font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-themecolor: accent2; mso-themeshade: 128; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi;">Kevin ||

media type="custom" key="3957319"media type="file" key="01 Track 01.m4a"

<span style="font-size: 12pt; color: #943634; font-family: 'Constantia','serif'; mso-themecolor: accent2; mso-themeshade: 191;">“Ain’t Nothin but an E-quation” is an inspirational song designed for the teaching of solving for zeros of polynomials. This song is recorded by Kdub and features Larry Da K.

<span style="color: #943634; font-family: Calibri; msothemecolor: accent2; msothemeshade: 191; mso-themecolor: accent2; mso-themeshade: 191;"> This section describes the Fundamental Theorem of Algebra, which states that in the complex number system, every nth-degree polynomial function has precisely n zeros. These zeros can be solved through the use of the Rational Zeros Test, which relates possible rational zeros of a polynomial (having leading coefficients) to the leading coefficient and the constant term of the polynomial. Another way in which one can solve for the zeros of a function is through factorization. Through factorization one finds the zeros of a function by solving for the linear and/or quadratic factors. Yet another way for one to discover the zeros of a function is through Descartes’ Rule of Signs in which one uses the variations in signs throughout a function to learn the number of positive or negative zeros derive from that function. <span style="display: block; font-size: 12pt; color: #632423; font-family: 'Constantia','serif'; text-align: center; msothemecolor: accent2; msothemeshade: 128;"> The content contained in this section is displayed in a very professional, confusing manner. In order to help you to remember the different concepts of this section we use sets of memorable pictures that spell these confusing concepts out!

<span style="font-size: 12pt; color: #d99594; font-family: 'Constantia','serif'; mso-themecolor: accent2; mso-themetint: 153;">We hope that you benefit from the use of this interactive website focused on the factorization of polynomials.

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<span style="font-size: 12pt; color: #5f497a; font-family: 'Constantia','serif'; mso-themecolor: accent4; mso-themeshade: 191;">So that you may fully understand all the concepts presented in this section, we have a PowerPoint presentation, equipped with step-by-step directions through each polynomial problem. We also have developed a glossary so that you can understand any vocabulary with which you are unfamiliar.