Chapter 3 Total Lecture Time: 03:28:58

Study Schedule = 13.5 Hours

You will see the belly of the ''polynomial beast'' in this chapter.  Don't even know what a polynomial is?  Don't be scared - we will prepare you well for your journey.  We will start by working with some of the friendlier polynomials, like quadratic equations, which you probably covered in Algebra 1.  (Don't worry, we will help you review.)  We will then expand our work to tougher (bigger) polynomials.  You will understand polynomials' behavior, you will learn how to divide them, you will learn to find their roots, and you will even predict how many roots there will be - almost as cool as predicting the future, and much easier.  Finally, we will learn how to graph them. Let's get started!


References and homework exercises are from College Algebra, 9th Edition, by Lial, Hornsby, and Schneider. ISBN 0-321-22757-3.
Answers and partial solutions to odd-numbered exercises are given in the back of the book.

Section 1 Total Lecture Time = 01:26:08

Study Schedule = 4. Hours

What is a polynomial?  If you've paid attention in English class (or maybe even Greek class), you might be able to figure this out. If you look at the roots, the word translates to "multiple terms."  Unsurprisingly, this is almost exactly what a polynomial is.

Then, what is a degree?  A piece of paper you get for graduating from somewhere?  Actually yes, but that's not what we're discussing here.  We will learn how to classify polynomials by degree, then we will review quadratic polynomials - a specific type of polynomial that has a degree of 2.  All quadratics can be described (extremely generally) as Ax2 + Bx + C.  How do these coefficients affect our functions?  And is there a more useful form?

Afterwards, we will learn how to rearrange and solve quadratic equations.  Specifically, we will talk about completing the square, by far the coolest way to solve a quadratic equation. (The quadratic formula is just way too boring.)  You should remember this from Algebra 1, but just in case you don't....  Finally, we will apply quadratic equations to solve real world problems of projectile motion and area/volume calculation.

3.1.1 Definition of Polynomial
Lecture Alg_2_03.01_01_DefinitionOfPolymial (00:03:45)
Reference:
College Algebra: Section 3.1.0 Polynomial Functions; Pg. 294
3.1.2 Degree of Polynomial Functions
Lecture Alg_2_03.01_02_DegreeOfPolynomialFunction (00:02:54)
Reference:
College Algebra: Section 3.1.0 Polynomial Functions; Pg. 294
3.1.3 Why We Don't Study the Zero and First Degree Polynomials
Lecture Alg_2_03.01_03_WhyWeDontStudyZeroAndFirstDegree (00:02:58)
Reference:
College Algebra: Section 3.1.0 Polynomial Functions; Pg. 294
3.1.4 Introduction to Quadratic Functions
Lecture Alg_2_03.01_04_IntroToQuadraticFunction (00:02:38)
Reference:
College Algebra: Section 3.1.1 Quadratic Functions; Pg. 294
3.1.5 The role of in
Lecture Alg_2_03.01_05_TheRoleOfA (00:03:56)
Reference:
College Algebra: Section 3.1.2 Graphing Techniques; Pg. 295
3.1.6 The role of and in
Lecture Alg_2_03.01_06_TheRoleOfBandC (00:02:39)
Reference:
College Algebra: Section 3.1.2 Graphing Techniques; Pg. 295
3.1.7 The and Form:
Lecture Alg_2_03.01_07_TheHandKForm (00:03:35)
Reference:
College Algebra: Section 3.1.2 Graphing Techniques; Example 1; Pg. 295
3.1.8 Two Ways to Write Quadratic Functions
Lecture Alg_2_03.01_08_TwoWaysToWriteQuadraticFunctions (00:01:16)
Reference:
College Algebra: Section 3.1.3 Completing the Square; Pg. 297
3.1.9 Completing the Square - Example 1
Lecture Alg_2_03.01_09_CompletingTheSquare1 (00:03:11)
Reference:
College Algebra: Section 3.1.3 Completing the Square; Pg. 297
3.1.10 Completing the Square - Example 2
Lecture Alg_2_03.01_10_CompletingTheSquare2 (00:01:23)
Reference:
College Algebra: Section 3.1.3 Completing the Square; Pg. 297
3.1.11 Completing the Square - Example 3
Lecture Alg_2_03.01_11_CompletingTheSquare3 (00:02:21)
Reference:
College Algebra: Section 3.1.3 Completing the Square; Pg. 297
3.1.12 Completing the Square - Example 4
Lecture Alg_2_03.01_12_CompletingTheSquare4 (00:03:52)
Reference:
College Algebra: Section 3.1.3 Completing the Square; Pg. 297
3.1.13 Completing the Square - Example 5
Lecture Alg_2_03.01_13_CompletingTheSquare5 (00:02:11)
Reference:
College Algebra: Section 3.1.3 Completing the Square; Pg. 297
3.1.14 Ultimate Completing the Square
Lecture Alg_2_03.01_14_UnltimateCompletingTheSquare (00:04:52)
Reference:
College Algebra: Section 3.1.3 Completing the Square; Examples 2 and 3; Pg. 297
3.1.15 Finding Vertex
Lecture Alg_2_03.01_15_FindingVertex (00:05:01)
Reference:
College Algebra: Section 3.1.4 The Vertex Formula; Example 4; Pg. 299
3.1.16.1 Introduction to Projectile Problems
Lecture Alg_2_03.01_16.1_ProjectileProblemIntro (00:07:53)
Reference:
College Algebra: Section 3.1.5 Quadratic Models and Curve Fitting; Pg. 300
3.1.16.2 Projectile Problems - Example 1
Lecture Alg_2_03.01_16.2_ProjectileProblemA (00:02:25)
Reference:
College Algebra: Section 3.1.5 Quadratic Models and Curve Fitting; Pg. 300
3.1.16.3 Projectile Problems - Example 2
Lecture Alg_2_03.01_16.3_ProjectileProblemB (00:03:10)
Reference:
College Algebra: Section 3.1.5 Quadratic Models and Curve Fitting; Pg. 300
3.1.16.4 Projectile Problems - Example 3
Lecture Alg_2_03.01_16.4_ProjectileProblemC (00:12:47)
Reference:
College Algebra: Section 3.1.5 Quadratic Models and Curve Fitting; Pg. 300
3.1.16.5 Projectile Problems - Example 4
Lecture Alg_2_03.01_16.5_ProjectileProblemD (00:03:32)
Reference:
College Algebra: Section 3.1.5 Quadratic Models and Curve Fitting; Example 5, Pg. 300
3.1.17 Box Problems
Lecture Alg_2_03.01_17_BoxProblem (00:09:39)
Reference:
College Algebra: Section 3.1.5 Quadratic Models and Curve Fitting; Pg. 300
Homework:
College Algebra: Section 3.1 Quadratic Functions;, Exercises 1~38 and 47~60

Section 2 Total Lecture Time = 00:28:31

Study Schedule = 2. Hours

You already know how to add, subtract, and multiply polynomials - but do you know how to divide them?  You might have learned polynomial long division in Algebra 1, but you probably haven't seen synthetic division.  It's like long division, except better and faster... but before we get there, let's review long division.  Believe it or not, all of these processes work just like the kind of long division you've been doing for years and years.

3.2.1 Meaning of Synthetic
Lecture Alg_2_03.02_01_MeaningOfSynthetic (00:00:49)
Reference:
College Algebra: Section 3.2.0 Division Algorithm; Pg. 313
3.2.2 Division Algorithm
Lecture Alg_2_03.02_02_DivisionAlgorithm (00:07:11)
Reference:
College Algebra: Section 3.2.0 Division Algorithm; Pg. 313
3.2.3 Synthetic Division by Long Hand
Lecture Alg_2_03.02_03_SyntheticDivisionLongHand (00:06:49)
Reference:
College Algebra: Section 3.2.1 Synthetic Division; Pg. 314
3.2.4 Synthetic Division
Lecture Alg_2_03.02_04_SyntheticDivision (00:08:57)
Reference:
College Algebra: Section 3.2.1 Synthetic Division; Pg. 314
3.2.5 Synthetic Division - Example 1
Lecture Alg_2_03.02_05_SyntheticDivisionExample1 (00:02:37)
Reference:
College Algebra: Section 3.2.1 Synthetic Division; Example 1; Pg. 314
3.2.6 Synthetic Division - Example 2
Lecture Alg_2_03.02_06_SyntheticDivisionExample2 (00:02:05)
Reference:
College Algebra: Section 3.2.1 Synthetic Division; Pg. 314
Homework:
College Algebra: Section 3.2 Synthetic Division; Exercises 1~26

Section 3 Total Lecture Time = 00:10:46

Study Schedule = 0.5 Hours

So now you know how to divide - we should be done, right?  Not exactly.  Remember those numbers we practically threw away after dividing, you know, remainders?  Remainders are actually the key to learning more about polynomials - who would have expected that?

3.3.1 Introduction to Remainder Theorem
Lecture Alg_2_03.03_01_RemainderTheorem (00:06:23)
Reference:
College Algebra: Section 3.2.2 Evaluating Polynomial Functions Using the Remainder Theorem; Pg. 316
3.3.2 Using Remainder Theorem - Example 1
Lecture Alg_2_03.03_02_UsingRemainderTheoremExample1 (00:02:07)
Reference:
College Algebra: Section 3.2.2 Evaluating Polynomial Functions Using the Remainder Theorem; Example 2; Pg. 316
College Algebra: Section 3.2.3 Testing Potential Zeros; Pg. 317
3.3.3 Using Remainder Theorem - Example 2
Lecture Alg_2_03.03_03_UsingRemainderTheoremExample2 (00:02:15)
Reference:
College Algebra: Section 3.2.3 Testing Potential Zeros; Example 3; Pg. 317
Homework:
College Algebra: Section 3.2 Synthetic Division; Exercises 27~56

Section 4 Total Lecture Time = 00:29:09

Study Schedule = 3. Hours

Remainders come up big again in the Factor Theorem.  You might have a hard time understanding the point of this theorem... but just have faith, we're building to some pretty interesting results here.

After mastering the Factor Theorem, you will be able to discern all the potential rational roots just by looking at a polynomial.  Unfortunately, the only way to know for sure is to test these numbers... but this is definitely an improvement over just guessing.


3.4.1 Introduction to Factor Theorem
Lecture Alg_2_03.04_01_FactorTheoremIntro (00:03:15)
Reference:
College Algebra: Section 3.3.1 Introduction to Factor Theorem; Pg. 320
3.4.2 An Illustration of Factor Theorem
Lecture Alg_2_03.04_02_FactorTheoremIllustration (00:07:37)
Reference:
College Algebra: Section 3.3.1 Introduction to Factor Theorem; Example 1 and 2; Pg. 320
3.4.3 Factor Theorem in Mathematica
Lecture Alg_2_03.04_03_FactorTheoremMathematica (00:06:29)
Reference:
College Algebra: Section 3.3.1 Introduction to Factor Theorem; Pg. 320
3.4.4 Rational Zero Theorem
Lecture Alg_2_03.04_04_RationalZeroTheorem (00:09:37)
Reference:
College Algebra: Section 3.3.2 Rational Zeros Theorem; Example 3; Pg. 322
3.4.5 Rational Zero Theorem in Mathematica
Lecture Alg_2_03.04_05_RationalZeroTheoremMathematica (00:02:08)
Reference:
College Algebra: Section 3.3.2 Rational Zeros Theorem; Pg. 322
Homework:
College Algebra: Section 3.3 Zeros of Polynomial Functions; Exercises 1~28 and 35~42

Section 5 Total Lecture Time = 00:16:19

Study Schedule = 1. Hours

This is a biggie - the Fundamental Theorem of Algebra. With that name, it better be important, right?  Although you might not see what the big deal is at first, trust me, it's definitely fundamental.  It also leads to a number of great other theorems.  How many zeros can a polynomial have?  What kind of numbers are these zeros?  Now you can find out the answers to these great life mysteries.

3.5.1 Fundamental Theorem of Algebra / Number of Zeros Theorem
Lecture Alg_2_03.05_01_FundamentalAlgebraNumberOfZeroTheorem (00:03:11)
Reference:
College Algebra: Section 3.3.3 Number of Zeros; Example 4; Pg. 324
3.5.2 Conjugate Zero Theorem
Lecture Alg_2_03.05_02_ConjugageZeroTheorem (00:06:33)
Reference:
College Algebra: Section 3.3.4 Conjugate Zeros Theorem; Example 5; Pg. 325
College Algebra: Section 3.3.5 Finding Zeros of a Polynomial Function; Example 6; Pg. 327
3.5.3 Descartes' Rule of Signs
Lecture Alg_2_03.05_03_DecartsRuleOfSigns (00:05:40)
Reference:
College Algebra: Section 3.3.6 Descartes' Rule of Signs; Pg. 328
3.5.4 Descartes' Rule of Signs - an Example
Lecture Alg_2_03.05_04_DecartsRuleOfSignsExample (00:00:53)
Reference:
College Algebra: Section 3.3.6 Descartes' Rule of Signs; Example 7; Pg. 328
Homework:
College Algebra: Section 3.3 Zeros of Polynomial Functions; Exercises 29,34 and 43~82

Section 6 Total Lecture Time = 00:38:04

Study Schedule = 3. Hours

So, now that we know everything about polynomials, let's check out what we can do with graphs.  Let's hope your Mathematica skills are up to par, because you're going to need them.  First, we will discuss how certain numbers in certain places can ''stretch,'' ''shrink,'' and ''shift'' polynomials.  Then we investigate the properties of odd and even polynomials.  What do these characteristics bring to the graphs?  We will also examine how to find zeros based on graphs.  Can you believe that we can even find where the graphs turn based just on the functions?  We can even figure out how they behave as they go off into the horizons (negative infinity or infinity).  Then we combine all the skills of this section to graph polynomial functions.

3.6.1 Graphs of
Lecture Alg_2_03.06_01_ax^nGraph (00:04:35)
Reference:
College Algebra: Section 3.4.1 Graphs of ; Example 1; Pg. 331
3.6.2 Graphs of
Lecture Alg_2_03.06_02_a(x-h)plusKGraph (00:02:13)
Reference:
College Algebra: Section 3.4.2 Graphs of General Polynomial Functions; Example 2; Pg. 332
3.6.3 Odd and Even Polynomials
Lecture Alg_2_03.06_03_OddAndEvenPolynomials (00:05:34)
Reference:
College Algebra: Section 3.4.2 Graphs of General Polynomial Functions; Pg. 332
3.6.4 Zeros of Polynomials
Lecture Alg_2_03.06_04_ZerosOfPolynomials (00:02:27)
Reference:
College Algebra: Section 3.4.2 Graphs of General Polynomial Functions; Pg. 332
3.6.5 Zeros of Polynomials in Mathematica
Lecture Alg_2_03.06_05_ZerosOfPolynomialsMathematica (00:06:04)
Reference:
College Algebra: Section 3.4.2 Graphs of General Polynomial Functions; Pg. 332
3.6.6 Turning Points
Lecture Alg_2_03.06_06_TurningPoints (00:05:18)
Reference:
College Algebra: Section 3.4.3 Turning Points and End Behavior; Pg. 334
3.6.7 End Behavior
Lecture Alg_2_03.06_07_EndBehavior (00:01:52)
Reference:
College Algebra: Section 3.4.3 Turning Points and End Behavior; Example 3; Pg. 334
3.6.8 Graphing Techniques for Polynomial Functions
Lecture Alg_2_03.06_08_PlotPolynomialFunction (00:04:27)
Reference:
College Algebra: Section 3.4.4 Graphing Techniques; Example 4; Pg. 336
3.6.9 Creating Polynomials from a Graph
Lecture Alg_2_03.06_09_CreatingPolynomialFromGraph (00:05:29)
Reference:
College Algebra: Section 3.4.7 Polynomial Models and Curve Fitting; Example 8; Pg. 341
Homework:
College Algebra: Section 3.4 Polynomials Functions: Graphs, Applications, and Models; Exercises 1~42