
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 0321227573.
Answers and partial solutions to oddnumbered 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 Ax^{2} + 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(xh)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
