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Start learning today, and be successful in your academic & professional career. Start Today!  Vincent Selhorst-Jones

Roots (Zeros) of Polynomials

Slide Duration:

Section 1: Introduction
Introduction to Precalculus

10m 3s

Intro
0:00
Title of the Course
0:06
Different Names for the Course
0:07
Precalculus
0:12
Math Analysis
0:14
Trigonometry
0:16
Algebra III
0:20
Geometry II
0:24
College Algebra
0:30
Same Concepts
0:36
How do the Lessons Work?
0:54
Introducing Concepts
0:56
Apply Concepts
1:04
Go through Examples
1:25
Who is this Course For?
1:38
Those Who Need eExtra Help with Class Work
1:52
Those Working on Material but not in Formal Class at School
1:54
Those Who Want a Refresher
2:00
Try to Watch the Whole Lesson
2:20
Understanding is So Important
3:56
What to Watch First
5:26
Lesson #2: Sets, Elements, and Numbers
5:30
Lesson #7: Idea of a Function
5:33
Lesson #6: Word Problems
6:04
What to Watch First, cont.
6:46
Lesson #2: Sets, Elements and Numbers
6:56
Lesson #3: Variables, Equations, and Algebra
6:58
Lesson #4: Coordinate Systems
7:00
Lesson #5: Midpoint, Distance, the Pythagorean Theorem and Slope
7:02
Lesson #6: Word Problems
7:10
Lesson #7: Idea of a Function
7:12
Lesson #8: Graphs
7:14
Graphing Calculator Appendix
7:40
What to Watch Last
8:46
Let's get Started!
9:48
Sets, Elements, & Numbers

45m 11s

Intro
0:00
Introduction
0:05
Sets and Elements
1:19
Set
1:20
Element
1:23
Name a Set
2:20
Order The Elements Appear In Has No Effect on the Set
2:55
Describing/ Defining Sets
3:28
Directly Say All the Elements
3:36
Clearly Describing All the Members of the Set
3:55
Describing the Quality (or Qualities) Each member Of the Set Has In Common
4:32
Symbols: 'Element of' and 'Subset of'
6:01
Symbol is ∈
6:03
Subset Symbol is ⊂
6:35
Empty Set
8:07
Symbol is ∅
8:20
Since It's Empty, It is a Subset of All Sets
8:44
Union and Intersection
9:54
Union Symbol is ∪
10:08
Intersection Symbol is ∩
10:18
Sets Can Be Weird Stuff
12:26
Can Have Elements in a Set
12:50
We Can Have Infinite Sets
13:09
Example
13:22
Consider a Set Where We Take a Word and Then Repeat It An Ever Increasing Number of Times
14:08
This Set Has Infinitely Many Distinct Elements
14:40
Numbers as Sets
16:03
Natural Numbers ℕ
16:16
Including 0 and the Negatives ℤ
18:13
Rational Numbers ℚ
19:27
Can Express Rational Numbers with Decimal Expansions
22:05
Irrational Numbers
23:37
Real Numbers ℝ: Put the Rational and Irrational Numbers Together
25:15
Interval Notation and the Real Numbers
26:45
Include the End Numbers
27:06
Exclude the End Numbers
27:33
Example
28:28
Interval Notation: Infinity
29:09
Use -∞ or ∞ to Show an Interval Going on Forever in One Direction or the Other
29:14
Always Use Parentheses
29:50
Examples
30:27
Example 1
31:23
Example 2
35:26
Example 3
38:02
Example 4
42:21
Variables, Equations, & Algebra

35m 31s

Intro
0:00
What is a Variable?
0:05
A Variable is a Placeholder for a Number
0:11
Affects the Output of a Function or a Dependent Variable
0:24
Naming Variables
1:51
Useful to Use Symbols
2:21
What is a Constant?
4:14
A Constant is a Fixed, Unchanging Number
4:28
We Might Refer to a Symbol Representing a Number as a Constant
4:51
What is a Coefficient?
5:33
A Coefficient is a Multiplicative Factor on a Variable
5:37
Not All Coefficients are Constants
5:51
Expressions and Equations
6:42
An Expression is a String of Mathematical Symbols That Make Sense Used Together
7:05
An Equation is a Statement That Two Expression Have the Same Value
8:20
The Idea of Algebra
8:51
Equality
8:59
If Two Things Are the Same *Equal), Then We Can Do the Exact Same Operation to Both and the Results Will Be the Same
9:41
Always Do The Exact Same Thing to Both Sides
12:22
Solving Equations
13:23
When You Are Asked to Solve an Equation, You Are Being Asked to Solve for Something
13:33
Look For What Values Makes the Equation True
13:38
Isolate the Variable by Doing Algebra
14:37
Order of Operations
16:02
Why Certain Operations are Grouped
17:01
When You Don't Have to Worry About Order
17:39
Distributive Property
18:15
It Allows Multiplication to Act Over Addition in Parentheses
18:23
We Can Use the Distributive Property in Reverse to Combine Like Terms
19:05
Substitution
20:03
Use Information From One Equation in Another Equation
20:07
20:44
Example 1
23:17
Example 2
25:49
Example 3
28:11
Example 4
30:02
Coordinate Systems

35m 2s

Intro
0:00
Inherent Order in ℝ
0:05
Real Numbers Come with an Inherent Order
0:11
Positive Numbers
0:21
Negative Numbers
0:58
'Less Than' and 'Greater Than'
2:04
2:56
Inequality
4:06
Less Than or Equal and Greater Than or Equal
4:51
One Dimension: The Number Line
5:36
Graphically Represent ℝ on a Number Line
5:43
Note on Infinities
5:57
With the Number Line, We Can Directly See the Order We Put on ℝ
6:35
Ordered Pairs
7:22
Example
7:34
Allows Us to Talk About Two Numbers at the Same Time
9:41
Ordered Pairs of Real Numbers Cannot be Put Into an Order Like we Did with ℝ
10:41
Two Dimensions: The Plane
13:13
We Can Represent Ordered Pairs with the Plane
13:24
Intersection is known as the Origin
14:31
Plotting the Point
14:32
Plane = Coordinate Plane = Cartesian Plane = ℝ²
17:46
18:50
19:04
19:21
20:04
20:20
Three Dimensions: Space
21:02
Create Ordered Triplets
21:09
Visually Represent This
21:19
Three-Dimension = Space = ℝ³
21:47
Higher Dimensions
22:24
If We Have n Dimensions, We Call It n-Dimensional Space or ℝ to the nth Power
22:31
We Can Represent Places In This n-Dimensional Space As Ordered Groupings of n Numbers
22:41
Hard to Visualize Higher Dimensional Spaces
23:18
Example 1
25:07
Example 2
26:10
Example 3
28:58
Example 4
31:05
Midpoints, Distance, the Pythagorean Theorem, & Slope

48m 43s

Intro
0:00
Introduction
0:07
Midpoint: One Dimension
2:09
Example of Something More Complex
2:31
Use the Idea of a Middle
3:28
Find the Midpoint of Arbitrary Values a and b
4:17
How They're Equivalent
5:05
Official Midpoint Formula
5:46
Midpoint: Two Dimensions
6:19
The Midpoint Must Occur at the Horizontal Middle and the Vertical Middle
6:38
Arbitrary Pair of Points Example
7:25
Distance: One Dimension
9:26
Absolute Value
10:54
Idea of Forcing Positive
11:06
Distance: One Dimension, Formula
11:47
Distance Between Arbitrary a and b
11:48
Absolute Value Helps When the Distance is Negative
12:41
Distance Formula
12:58
The Pythagorean Theorem
13:24
a²+b²=c²
13:50
Distance: Two Dimensions
14:59
Break Into Horizontal and Vertical Parts and then Use the Pythagorean Theorem
15:16
Distance Between Arbitrary Points (x₁,y₁) and (x₂,y₂)
16:21
Slope
19:30
Slope is the Rate of Change
19:41
m = rise over run
21:27
Slope Between Arbitrary Points (x₁,y₁) and (x₂,y₂)
22:31
Interpreting Slope
24:12
Positive Slope and Negative Slope
25:40
m=1, m=0, m=-1
26:48
Example 1
28:25
Example 2
31:42
Example 3
36:40
Example 4
42:48
Word Problems

56m 31s

Intro
0:00
Introduction
0:05
What is a Word Problem?
0:45
Describes Any Problem That Primarily Gets Its Ideas Across With Words Instead of Math Symbols
0:48
Requires Us to Think
1:32
Why Are They So Hard?
2:11
Reason 1: No Simple Formula to Solve Them
2:16
Reason 2: Harder to Teach Word Problems
2:47
You Can Learn How to Do Them!
3:51
7:57
'But I'm Never Going to Use This In Real Life'
9:46
Solving Word Problems
12:58
First: Understand the Problem
13:37
Second: What Are You Looking For?
14:33
Third: Set Up Relationships
16:21
Fourth: Solve It!
17:48
Summary of Method
19:04
Examples on Things Other Than Math
20:21
Math-Specific Method: What You Need Now
25:30
Understand What the Problem is Talking About
25:37
Set Up and Name Any Variables You Need to Know
25:56
Set Up Equations Connecting Those Variables to the Information in the Problem Statement
26:02
Use the Equations to Solve for an Answer
26:14
Tip
26:58
Draw Pictures
27:22
Breaking Into Pieces
28:28
Try Out Hypothetical Numbers
29:52
Student Logic
31:27
Jump In!
32:40
Example 1
34:03
Example 2
39:15
Example 3
44:22
Example 4
50:24
Section 2: Functions
Idea of a Function

39m 54s

Intro
0:00
Introduction
0:04
What is a Function?
1:06
A Visual Example and Non-Example
1:30
Function Notation
3:47
f(x)
4:05
Express What Sets the Function Acts On
5:45
Metaphors for a Function
6:17
Transformation
6:28
Map
7:17
Machine
8:56
Same Input Always Gives Same Output
10:01
If We Put the Same Input Into a Function, It Will Always Produce the Same Output
10:11
Example of Something That is Not a Function
11:10
A Non-Numerical Example
12:10
The Functions We Will Use
15:05
Unless Told Otherwise, We Will Assume Every Function Takes in Real Numbers and Outputs Real Numbers
15:11
Usually Told the Rule of a Given Function
15:27
How To Use a Function
16:18
Apply the Rule to Whatever Our Input Value Is
16:28
Make Sure to Wrap Your Substitutions in Parentheses
17:09
Functions and Tables
17:36
Table of Values, Sometimes Called a T-Table
17:46
Example
17:56
Domain: What Goes In
18:55
The Domain is the Set of all Inputs That the Function Can Accept
18:56
Example
19:40
Range: What Comes Out
21:27
The Range is the Set of All Possible Outputs a Function Can Assign
21:34
Example
21:49
Another Example Would Be Our Initial Function From Earlier in This Lesson
22:29
Example 1
23:45
Example 2
25:22
Example 3
27:27
Example 4
29:23
Example 5
33:33
Graphs

58m 26s

Intro
0:00
Introduction
0:04
How to Interpret Graphs
1:17
Input / Independent Variable
1:47
Output / Dependent Variable
2:00
Graph as Input ⇒ Output
2:23
One Way to Think of a Graph: See What Happened to Various Inputs
2:25
Example
2:47
Graph as Location of Solution
4:20
A Way to See Solutions
4:36
Example
5:20
Which Way Should We Interpret?
7:13
Easiest to Think In Terms of How Inputs Are Mapped to Outputs
7:20
Sometimes It's Easier to Think In Terms of Solutions
8:39
Pay Attention to Axes
9:50
Axes Tell Where the Graph Is and What Scale It Has
10:09
Often, The Axes Will Be Square
10:14
Example
12:06
Arrows or No Arrows?
16:07
Will Not Use Arrows at the End of Our Graphs
17:13
Graph Stops Because It Hits the Edge of the Graphing Axes, Not Because the Function Stops
17:18
How to Graph
19:47
Plot Points
20:07
Connect with Curves
21:09
If You Connect with Straight Lines
21:44
Graphs of Functions are Smooth
22:21
More Points ⇒ More Accurate
23:38
Vertical Line Test
27:44
If a Vertical Line Could Intersect More Than One Point On a Graph, It Can Not Be the Graph of a Function
28:41
Every Point on a Graph Tells Us Where the x-Value Below is Mapped
30:07
Domain in Graphs
31:37
The Domain is the Set of All Inputs That a Function Can Accept
31:44
Be Aware That Our Function Probably Continues Past the Edge of Our 'Viewing Window'
33:19
Range in Graphs
33:53
Graphing Calculators: Check the Appendix!
36:55
Example 1
38:37
Example 2
45:19
Example 3
50:41
Example 4
53:28
Example 5
55:50
Properties of Functions

48m 49s

Intro
0:00
Introduction
0:05
Increasing Decreasing Constant
0:43
Looking at a Specific Graph
1:15
Increasing Interval
2:39
Constant Function
4:15
Decreasing Interval
5:10
Find Intervals by Looking at the Graph
5:32
Intervals Show x-values; Write in Parentheses
6:39
Maximum and Minimums
8:48
Relative (Local) Max/Min
10:20
Formal Definition of Relative Maximum
12:44
Formal Definition of Relative Minimum
13:05
Max/Min, More Terms
14:18
Definition of Extrema
15:01
Average Rate of Change
16:11
Drawing a Line for the Average Rate
16:48
Using the Slope of the Secant Line
17:36
Slope in Function Notation
18:45
Zeros/Roots/x-intercepts
19:45
What Zeros in a Function Mean
20:25
Even Functions
22:30
Odd Functions
24:36
Even/Odd Functions and Graphs
26:28
Example of an Even Function
27:12
Example of an Odd Function
28:03
Example 1
29:35
Example 2
33:07
Example 3
40:32
Example 4
42:34
Function Petting Zoo

29m 20s

Intro
0:00
Introduction
0:04
Don't Forget that Axes Matter!
1:44
The Constant Function
2:40
The Identity Function
3:44
The Square Function
4:40
The Cube Function
5:44
The Square Root Function
6:51
The Reciprocal Function
8:11
The Absolute Value Function
10:19
The Trigonometric Functions
11:56
f(x)=sin(x)
12:12
f(x)=cos(x)
12:24
Alternate Axes
12:40
The Exponential and Logarithmic Functions
13:35
Exponential Functions
13:44
Logarithmic Functions
14:24
Alternating Axes
15:17
Transformations and Compositions
16:08
Example 1
17:52
Example 2
18:33
Example 3
20:24
Example 4
26:07
Transformation of Functions

48m 35s

Intro
0:00
Introduction
0:04
Vertical Shift
1:12
Graphical Example
1:21
A Further Explanation
2:16
Vertical Stretch/Shrink
3:34
Graph Shrinks
3:46
Graph Stretches
3:51
A Further Explanation
5:07
Horizontal Shift
6:49
Moving the Graph to the Right
7:28
Moving the Graph to the Left
8:12
A Further Explanation
8:19
Understanding Movement on the x-axis
8:38
Horizontal Stretch/Shrink
12:59
Shrinking the Graph
13:40
Stretching the Graph
13:48
A Further Explanation
13:55
Understanding Stretches from the x-axis
14:12
Vertical Flip (aka Mirror)
16:55
Example Graph
17:07
Multiplying the Vertical Component by -1
17:18
Horizontal Flip (aka Mirror)
18:43
Example Graph
19:01
Multiplying the Horizontal Component by -1
19:54
Summary of Transformations
22:11
Stacking Transformations
24:46
Order Matters
25:20
Transformation Example
25:52
Example 1
29:21
Example 2
34:44
Example 3
38:10
Example 4
43:46
Composite Functions

33m 24s

Intro
0:00
Introduction
0:04
Arithmetic Combinations
0:40
Basic Operations
1:20
Definition of the Four Arithmetic Combinations
1:40
Composite Functions
2:53
The Function as a Machine
3:32
Function Compositions as Multiple Machines
3:59
Notation for Composite Functions
4:46
Two Formats
6:02
Another Visual Interpretation
7:17
How to Use Composite Functions
8:21
Example of on Function acting on Another
9:17
Example 1
11:03
Example 2
15:27
Example 3
21:11
Example 4
27:06
Piecewise Functions

51m 42s

Intro
0:00
Introduction
0:04
Analogies to a Piecewise Function
1:16
Different Potatoes
1:41
Factory Production
2:27
Notations for Piecewise Functions
3:39
Notation Examples from Analogies
6:11
Example of a Piecewise (with Table)
7:24
Example of a Non-Numerical Piecewise
11:35
Graphing Piecewise Functions
14:15
Graphing Piecewise Functions, Example
16:26
Continuous Functions
16:57
Statements of Continuity
19:30
Example of Continuous and Non-Continuous Graphs
20:05
Interesting Functions: the Step Function
22:00
Notation for the Step Function
22:40
How the Step Function Works
22:56
Graph of the Step Function
25:30
Example 1
26:22
Example 2
28:49
Example 3
36:50
Example 4
46:11
Inverse Functions

49m 37s

Intro
0:00
Introduction
0:04
Analogy by picture
1:10
How to Denote the inverse
1:40
What Comes out of the Inverse
1:52
Requirement for Reversing
2:02
2:12
The Importance of Information
2:45
One-to-One
4:04
Requirement for Reversibility
4:21
When a Function has an Inverse
4:43
One-to-One
5:13
Not One-to-One
5:50
Not a Function
6:19
Horizontal Line Test
7:01
How to the test Works
7:12
One-to-One
8:12
Not One-to-One
8:45
Definition: Inverse Function
9:12
Formal Definition
9:21
Caution to Students
10:02
Domain and Range
11:12
Finding the Range of the Function Inverse
11:56
Finding the Domain of the Function Inverse
12:11
Inverse of an Inverse
13:09
Its just x!
13:26
Proof
14:03
Graphical Interpretation
17:07
Horizontal Line Test
17:20
Graph of the Inverse
18:04
Swapping Inputs and Outputs to Draw Inverses
19:02
How to Find the Inverse
21:03
What We Are Looking For
21:21
Reversing the Function
21:38
A Method to Find Inverses
22:33
Check Function is One-to-One
23:04
Swap f(x) for y
23:25
Interchange x and y
23:41
Solve for y
24:12
Replace y with the inverse
24:40
25:01
Keeping Step 2 and 3 Straight
25:44
Switching to Inverse
26:12
Checking Inverses
28:52
How to Check an Inverse
29:06
Quick Example of How to Check
29:56
Example 1
31:48
Example 2
34:56
Example 3
39:29
Example 4
46:19
Variation Direct and Inverse

28m 49s

Intro
0:00
Introduction
0:06
Direct Variation
1:14
Same Direction
1:21
Common Example: Groceries
1:56
Different Ways to Say that Two Things Vary Directly
2:28
Basic Equation for Direct Variation
2:55
Inverse Variation
3:40
Opposite Direction
3:50
Common Example: Gravity
4:53
Different Ways to Say that Two Things Vary Indirectly
5:48
Basic Equation for Indirect Variation
6:33
Joint Variation
7:27
Equation for Joint Variation
7:53
Explanation of the Constant
8:48
Combined Variation
9:35
Gas Law as a Combination
9:44
Single Constant
10:33
Example 1
10:49
Example 2
13:34
Example 3
15:39
Example 4
19:48
Section 3: Polynomials
Intro to Polynomials

38m 41s

Intro
0:00
Introduction
0:04
Definition of a Polynomial
1:04
Starting Integer
2:06
Structure of a Polynomial
2:49
The a Constants
3:34
Polynomial Function
5:13
Polynomial Equation
5:23
Polynomials with Different Variables
5:36
Degree
6:23
Informal Definition
6:31
Find the Largest Exponent Variable
6:44
Quick Examples
7:36
Special Names for Polynomials
8:59
Based on the Degree
9:23
Based on the Number of Terms
10:12
Distributive Property (aka 'FOIL')
11:37
Basic Distributive Property
12:21
Distributing Two Binomials
12:55
Longer Parentheses
15:12
Reverse: Factoring
17:26
Long-Term Behavior of Polynomials
17:48
Examples
18:13
Controlling Term--Term with the Largest Exponent
19:33
Positive and Negative Coefficients on the Controlling Term
20:21
22:07
Even Degree, Positive Coefficient
22:13
Even Degree, Negative Coefficient
22:39
Odd Degree, Positive Coefficient
23:09
Odd Degree, Negative Coefficient
23:27
Example 1
25:11
Example 2
27:16
Example 3
31:16
Example 4
34:41
Roots (Zeros) of Polynomials

41m 7s

Intro
0:00
Introduction
0:05
Roots in Graphs
1:17
The x-intercepts
1:33
How to Remember What 'Roots' Are
1:50
Naïve Attempts
2:31
Isolating Variables
2:45
Failures of Isolating Variables
3:30
Missing Solutions
4:59
Factoring: How to Find Roots
6:28
How Factoring Works
6:36
Why Factoring Works
7:20
Steps to Finding Polynomial Roots
9:21
Factoring: How to Find Roots CAUTION
10:08
Factoring is Not Easy
11:32
13:08
13:21
Form of Factored Binomials
13:38
Factoring Examples
14:40
16:58
Factoring Higher Degree Polynomials
18:19
Factoring a Cubic
18:32
19:04
Factoring: Roots Imply Factors
19:54
Where a Root is, A Factor Is
20:01
How to Use Known Roots to Make Factoring Easier
20:35
Not all Polynomials Can be Factored
22:30
Irreducible Polynomials
23:27
Complex Numbers Help
23:55
Max Number of Roots/Factors
24:57
Limit to Number of Roots Equal to the Degree
25:18
Why there is a Limit
25:25
Max Number of Peaks/Valleys
26:39
Shape Information from Degree
26:46
Example Graph
26:54
Max, But Not Required
28:00
Example 1
28:37
Example 2
31:21
Example 3
36:12
Example 4
38:40
Completing the Square and the Quadratic Formula

39m 43s

Intro
0:00
Introduction
0:05
Square Roots and Equations
0:51
Taking the Square Root to Find the Value of x
0:55
Getting the Positive and Negative Answers
1:05
Completing the Square: Motivation
2:04
Polynomials that are Easy to Solve
2:20
Making Complex Polynomials Easy to Solve
3:03
Steps to Completing the Square
4:30
Completing the Square: Method
7:22
Move C over
7:35
Divide by A
7:44
Find r
7:59
Add to Both Sides to Complete the Square
8:49
9:56
11:38
Derivation
11:43
Final Form
12:23
13:38
How Many Roots?
14:53
The Discriminant
15:47
What the Discriminant Tells Us: How Many Roots
15:58
How the Discriminant Works
16:30
Example 1: Complete the Square
18:24
22:00
Example 3: Solve for Zeroes
25:28
Example 4: Using the Quadratic Formula
30:52

45m 34s

Intro
0:00
Introduction
0:05
Parabolas
0:35
Examples of Different Parabolas
1:06
Axis of Symmetry and Vertex
1:28
Drawing an Axis of Symmetry
1:51
Placing the Vertex
2:28
Looking at the Axis of Symmetry and Vertex for other Parabolas
3:09
Transformations
4:18
Reviewing Transformation Rules
6:28
Note the Different Horizontal Shift Form
7:45
8:54
The Constants: k, h, a
9:05
Transformations Formed
10:01
Analyzing Different Parabolas
10:10
Switching Forms by Completing the Square
11:43
Vertex of a Parabola
16:30
Vertex at (h, k)
16:47
Vertex in Terms of a, b, and c Coefficients
17:28
Minimum/Maximum at Vertex
18:19
When a is Positive
18:25
When a is Negative
18:52
Axis of Symmetry
19:54
Incredibly Minor Note on Grammar
20:52
Example 1
21:48
Example 2
26:35
Example 3
28:55
Example 4
31:40
Intermediate Value Theorem and Polynomial Division

46m 8s

Intro
0:00
Introduction
0:05
Reminder: Roots Imply Factors
1:32
The Intermediate Value Theorem
3:41
The Basis: U between a and b
4:11
U is on the Function
4:52
Intermediate Value Theorem, Proof Sketch
5:51
If Not True, the Graph Would Have to Jump
5:58
But Graph is Defined as Continuous
6:43
Finding Roots with the Intermediate Value Theorem
7:01
Picking a and b to be of Different Signs
7:10
Must Be at Least One Root
7:46
Dividing a Polynomial
8:16
Using Roots and Division to Factor
8:38
Long Division Refresher
9:08
The Division Algorithm
12:18
How It Works to Divide Polynomials
12:37
The Parts of the Equation
13:24
Rewriting the Equation
14:47
Polynomial Long Division
16:20
Polynomial Long Division In Action
16:29
One Step at a Time
20:51
Synthetic Division
22:46
Setup
23:11
Synthetic Division, Example
24:44
Which Method Should We Use
26:39
26:49
27:13
Example 1
29:24
Example 2
31:27
Example 3
36:22
Example 4
40:55
Complex Numbers

45m 36s

Intro
0:00
Introduction
0:04
A Wacky Idea
1:02
The Definition of the Imaginary Number
1:22
How it Helps Solve Equations
2:20
Square Roots and Imaginary Numbers
3:15
Complex Numbers
5:00
Real Part and Imaginary Part
5:20
When Two Complex Numbers are Equal
6:10
6:40
Deal with Real and Imaginary Parts Separately
7:36
Two Quick Examples
7:54
Multiplication
9:07
FOIL Expansion
9:14
Note What Happens to the Square of the Imaginary Number
9:41
Two Quick Examples
10:22
Division
11:27
Complex Conjugates
13:37
Getting Rid of i
14:08
How to Denote the Conjugate
14:48
Division through Complex Conjugates
16:11
Multiply by the Conjugate of the Denominator
16:28
Example
17:46
19:24
20:12
Conjugate Pairs
20:37
But Are the Complex Numbers 'Real'?
21:27
What Makes a Number Legitimate
25:38
Where Complex Numbers are Used
27:20
Still, We Won't See Much of C
29:05
Example 1
30:30
Example 2
33:15
Example 3
38:12
Example 4
42:07
Fundamental Theorem of Algebra

19m 9s

Intro
0:00
Introduction
0:05
Idea: Hidden Roots
1:16
Roots in Complex Form
1:42
All Polynomials Have Roots
2:08
Fundamental Theorem of Algebra
2:21
Where Are All the Imaginary Roots, Then?
3:17
All Roots are Complex
3:45
Real Numbers are a Subset of Complex Numbers
3:59
The n Roots Theorem
5:01
For Any Polynomial, Its Degree is Equal to the Number of Roots
5:11
Equivalent Statement
5:24
6:29
Non-Distinct Roots
6:59
Denoting Multiplicity
7:20
7:41
8:55
9:59
Proof Sketch of n Roots Theorem
10:45
First Root
11:36
Second Root
13:23
Continuation to Find all Roots
16:00
Section 4: Rational Functions
Rational Functions and Vertical Asymptotes

33m 22s

Intro
0:00
Introduction
0:05
Definition of a Rational Function
1:20
Examples of Rational Functions
2:30
Why They are Called 'Rational'
2:47
Domain of a Rational Function
3:15
Undefined at Denominator Zeros
3:25
Otherwise all Reals
4:16
Investigating a Fundamental Function
4:50
The Domain of the Function
5:04
What Occurs at the Zeroes of the Denominator
5:20
Idea of a Vertical Asymptote
6:23
What's Going On?
6:58
Approaching x=0 from the left
7:32
Approaching x=0 from the right
8:34
Dividing by Very Small Numbers Results in Very Large Numbers
9:31
Definition of a Vertical Asymptote
10:05
Vertical Asymptotes and Graphs
11:15
Drawing Asymptotes by Using a Dashed Line
11:27
The Graph Can Never Touch Its Undefined Point
12:00
Not All Zeros Give Asymptotes
13:02
Special Cases: When Numerator and Denominator Go to Zero at the Same Time
14:58
Cancel out Common Factors
15:49
How to Find Vertical Asymptotes
16:10
Figure out What Values Are Not in the Domain of x
16:24
Determine if the Numerator and Denominator Share Common Factors and Cancel
16:45
Find Denominator Roots
17:33
Note if Asymptote Approaches Negative or Positive Infinity
18:06
Example 1
18:57
Example 2
21:26
Example 3
23:04
Example 4
30:01
Horizontal Asymptotes

34m 16s

Intro
0:00
Introduction
0:05
Investigating a Fundamental Function
0:53
What Happens as x Grows Large
1:00
Different View
1:12
Idea of a Horizontal Asymptote
1:36
What's Going On?
2:24
What Happens as x Grows to a Large Negative Number
2:49
What Happens as x Grows to a Large Number
3:30
Dividing by Very Large Numbers Results in Very Small Numbers
3:52
Example Function
4:41
Definition of a Vertical Asymptote
8:09
Expanding the Idea
9:03
What's Going On?
9:48
What Happens to the Function in the Long Run?
9:51
Rewriting the Function
10:13
Definition of a Slant Asymptote
12:09
Symbolical Definition
12:30
Informal Definition
12:45
Beyond Slant Asymptotes
13:03
Not Going Beyond Slant Asymptotes
14:39
Horizontal/Slant Asymptotes and Graphs
15:43
How to Find Horizontal and Slant Asymptotes
16:52
How to Find Horizontal Asymptotes
17:12
Expand the Given Polynomials
17:18
Compare the Degrees of the Numerator and Denominator
17:40
How to Find Slant Asymptotes
20:05
Slant Asymptotes Exist When n+m=1
20:08
Use Polynomial Division
20:24
Example 1
24:32
Example 2
25:53
Example 3
26:55
Example 4
29:22
Graphing Asymptotes in a Nutshell

49m 7s

Intro
0:00
Introduction
0:05
A Process for Graphing
1:22
1. Factor Numerator and Denominator
1:50
2. Find Domain
2:53
3. Simplifying the Function
3:59
4. Find Vertical Asymptotes
4:59
5. Find Horizontal/Slant Asymptotes
5:24
6. Find Intercepts
7:35
7. Draw Graph (Find Points as Necessary)
9:21
Draw Graph Example
11:21
Vertical Asymptote
11:41
Horizontal Asymptote
11:50
Other Graphing
12:16
Test Intervals
15:08
Example 1
17:57
Example 2
23:01
Example 3
29:02
Example 4
33:37
Partial Fractions

44m 56s

Intro
0:00
Introduction: Idea
0:04
Introduction: Prerequisites and Uses
1:57
Proper vs. Improper Polynomial Fractions
3:11
Possible Things in the Denominator
4:38
Linear Factors
6:16
Example of Linear Factors
7:03
Multiple Linear Factors
7:48
8:25
9:26
9:49
Mixing Factor Types
10:28
Figuring Out the Numerator
11:10
How to Solve for the Constants
11:30
Quick Example
11:40
Example 1
14:29
Example 2
18:35
Example 3
20:33
Example 4
28:51
Section 5: Exponential & Logarithmic Functions
Understanding Exponents

35m 17s

Intro
0:00
Introduction
0:05
Fundamental Idea
1:46
Expanding the Idea
2:28
Multiplication of the Same Base
2:40
Exponents acting on Exponents
3:45
Different Bases with the Same Exponent
4:31
To the Zero
5:35
To the First
5:45
Fundamental Rule with the Zero Power
6:35
To the Negative
7:45
Any Number to a Negative Power
8:14
A Fraction to a Negative Power
9:58
Division with Exponential Terms
10:41
To the Fraction
11:33
Square Root
11:58
Any Root
12:59
Summary of Rules
14:38
To the Irrational
17:21
Example 1
20:34
Example 2
23:42
Example 3
27:44
Example 4
31:44
Example 5
33:15
Exponential Functions

47m 4s

Intro
0:00
Introduction
0:05
Definition of an Exponential Function
0:48
Definition of the Base
1:02
Restrictions on the Base
1:16
Computing Exponential Functions
2:29
Harder Computations
3:10
When to Use a Calculator
3:21
Graphing Exponential Functions: a>1
6:02
Three Examples
6:13
What to Notice on the Graph
7:44
A Story
8:27
Story Diagram
9:15
Increasing Exponentials
11:29
Story Morals
14:40
Application: Compound Interest
15:15
Compounding Year after Year
16:01
Function for Compounding Interest
16:51
A Special Number: e
20:55
Expression for e
21:28
Where e stabilizes
21:55
Application: Continuously Compounded Interest
24:07
Equation for Continuous Compounding
24:22
Exponential Decay 0<a<1
25:50
Three Examples
26:11
Why they 'lose' value
26:54
Example 1
27:47
Example 2
33:11
Example 3
36:34
Example 4
41:28
Introduction to Logarithms

40m 31s

Intro
0:00
Introduction
0:04
Definition of a Logarithm, Base 2
0:51
Log 2 Defined
0:55
Examples
2:28
Definition of a Logarithm, General
3:23
Examples of Logarithms
5:15
Problems with Unusual Bases
7:38
Shorthand Notation: ln and log
9:44
base e as ln
10:01
base 10 as log
10:34
Calculating Logarithms
11:01
using a calculator
11:34
issues with other bases
11:58
Graphs of Logarithms
13:21
Three Examples
13:29
Slow Growth
15:19
Logarithms as Inverse of Exponentiation
16:02
Using Base 2
16:05
General Case
17:10
Looking More Closely at Logarithm Graphs
19:16
The Domain of Logarithms
20:41
21:08
The Alternate
24:00
Example 1
25:59
Example 2
30:03
Example 3
32:49
Example 4
37:34
Properties of Logarithms

42m 33s

Intro
0:00
Introduction
0:04
Basic Properties
1:12
Inverse--log(exp)
1:43
A Key Idea
2:44
What We Get through Exponentiation
3:18
B Always Exists
4:50
Inverse--exp(log)
5:53
Logarithm of a Power
7:44
Logarithm of a Product
10:07
Logarithm of a Quotient
13:48
Caution! There Is No Rule for loga(M+N)
16:12
Summary of Properties
17:42
Change of Base--Motivation
20:17
No Calculator Button
20:59
A Specific Example
21:45
Simplifying
23:45
Change of Base--Formula
24:14
Example 1
25:47
Example 2
29:08
Example 3
31:14
Example 4
34:13
Solving Exponential and Logarithmic Equations

34m 10s

Intro
0:00
Introduction
0:05
One to One Property
1:09
Exponential
1:26
Logarithmic
1:44
Specific Considerations
2:02
One-to-One Property
3:30
Solving by One-to-One
4:11
Inverse Property
6:09
Solving by Inverses
7:25
Dealing with Equations
7:50
Example of Taking an Exponent or Logarithm of an Equation
9:07
A Useful Property
11:57
Bring Down Exponents
12:01
Try to Simplify
13:20
Extraneous Solutions
13:45
Example 1
16:37
Example 2
19:39
Example 3
21:37
Example 4
26:45
Example 5
29:37
Application of Exponential and Logarithmic Functions

48m 46s

Intro
0:00
Introduction
0:06
Applications of Exponential Functions
1:07
A Secret!
2:17
Natural Exponential Growth Model
3:07
Figure out r
3:34
A Secret!--Why Does It Work?
4:44
e to the r Morphs
4:57
Example
5:06
Applications of Logarithmic Functions
8:32
Examples
8:43
What Logarithms are Useful For
9:53
Example 1
11:29
Example 2
15:30
Example 3
26:22
Example 4
32:05
Example 5
39:19
Section 6: Trigonometric Functions
Angles

39m 5s

Intro
0:00
Degrees
0:22
Circle is 360 Degrees
0:48
Splitting a Circle
1:13
2:08
2:31
2:52
Half-Circle and Right Angle
4:00
6:24
6:52
Coterminal, Complementary, Supplementary Angles
7:23
Coterminal Angles
7:30
Complementary Angles
9:40
Supplementary Angles
10:08
Example 1: Dividing a Circle
10:38
Example 2: Converting Between Degrees and Radians
11:56
Example 3: Quadrants and Coterminal Angles
14:18
Extra Example 1: Common Angle Conversions
-1
Extra Example 2: Quadrants and Coterminal Angles
-2
Sine and Cosine Functions

43m 16s

Intro
0:00
Sine and Cosine
0:15
Unit Circle
0:22
Coordinates on Unit Circle
1:03
Right Triangles
1:52
2:25
Master Right Triangle Formula: SOHCAHTOA
2:48
Odd Functions, Even Functions
4:40
Example: Odd Function
4:56
Example: Even Function
7:30
Example 1: Sine and Cosine
10:27
Example 2: Graphing Sine and Cosine Functions
14:39
Example 3: Right Triangle
21:40
Example 4: Odd, Even, or Neither
26:01
Extra Example 1: Right Triangle
-1
Extra Example 2: Graphing Sine and Cosine Functions
-2
Sine and Cosine Values of Special Angles

33m 5s

Intro
0:00
45-45-90 Triangle and 30-60-90 Triangle
0:08
45-45-90 Triangle
0:21
30-60-90 Triangle
2:06
Mnemonic: All Students Take Calculus (ASTC)
5:21
Using the Unit Circle
5:59
New Angles
6:21
9:43
Mnemonic: All Students Take Calculus
10:13
13:11
16:48
Example 3: All Angles and Quadrants
20:21
Extra Example 1: Convert, Quadrant, Sine/Cosine
-1
Extra Example 2: All Angles and Quadrants
-2
Modified Sine Waves: Asin(Bx+C)+D and Acos(Bx+C)+D

52m 3s

Intro
0:00
Amplitude and Period of a Sine Wave
0:38
Sine Wave Graph
0:58
Amplitude: Distance from Middle to Peak
1:18
Peak: Distance from Peak to Peak
2:41
Phase Shift and Vertical Shift
4:13
Phase Shift: Distance Shifted Horizontally
4:16
Vertical Shift: Distance Shifted Vertically
6:48
Example 1: Amplitude/Period/Phase and Vertical Shift
8:04
Example 2: Amplitude/Period/Phase and Vertical Shift
17:39
Example 3: Find Sine Wave Given Attributes
25:23
Extra Example 1: Amplitude/Period/Phase and Vertical Shift
-1
Extra Example 2: Find Cosine Wave Given Attributes
-2
Tangent and Cotangent Functions

36m 4s

Intro
0:00
Tangent and Cotangent Definitions
0:21
Tangent Definition
0:25
Cotangent Definition
0:47
Master Formula: SOHCAHTOA
1:01
Mnemonic
1:16
Tangent and Cotangent Values
2:29
Remember Common Values of Sine and Cosine
2:46
90 Degrees Undefined
4:36
Slope and Menmonic: ASTC
5:47
Uses of Tangent
5:54
Example: Tangent of Angle is Slope
6:09
7:49
Example 1: Graph Tangent and Cotangent Functions
10:42
Example 2: Tangent and Cotangent of Angles
16:09
Example 3: Odd, Even, or Neither
18:56
Extra Example 1: Tangent and Cotangent of Angles
-1
Extra Example 2: Tangent and Cotangent of Angles
-2
Secant and Cosecant Functions

27m 18s

Intro
0:00
Secant and Cosecant Definitions
0:17
Secant Definition
0:18
Cosecant Definition
0:33
Example 1: Graph Secant Function
0:48
Example 2: Values of Secant and Cosecant
6:49
Example 3: Odd, Even, or Neither
12:49
Extra Example 1: Graph of Cosecant Function
-1
Extra Example 2: Values of Secant and Cosecant
-2
Inverse Trigonometric Functions

32m 58s

Intro
0:00
Arcsine Function
0:24
Restrictions between -1 and 1
0:43
Arcsine Notation
1:26
Arccosine Function
3:07
Restrictions between -1 and 1
3:36
Cosine Notation
3:53
Arctangent Function
4:30
Between -Pi/2 and Pi/2
4:44
Tangent Notation
5:02
Example 1: Domain/Range/Graph of Arcsine
5:45
Example 2: Arcsin/Arccos/Arctan Values
10:46
Example 3: Domain/Range/Graph of Arctangent
17:14
Extra Example 1: Domain/Range/Graph of Arccosine
-1
Extra Example 2: Arcsin/Arccos/Arctan Values
-2
Computations of Inverse Trigonometric Functions

31m 8s

Intro
0:00
Inverse Trigonometric Function Domains and Ranges
0:31
Arcsine
0:41
Arccosine
1:14
Arctangent
1:41
Example 1: Arcsines of Common Values
2:44
Example 2: Odd, Even, or Neither
5:57
Example 3: Arccosines of Common Values
12:24
Extra Example 1: Arctangents of Common Values
-1
Extra Example 2: Arcsin/Arccos/Arctan Values
-2
Section 7: Trigonometric Identities
Pythagorean Identity

19m 11s

Intro
0:00
Pythagorean Identity
0:17
Pythagorean Triangle
0:27
Pythagorean Identity
0:45
Example 1: Use Pythagorean Theorem to Prove Pythagorean Identity
1:14
Example 2: Find Angle Given Cosine and Quadrant
4:18
Example 3: Verify Trigonometric Identity
8:00
Extra Example 1: Use Pythagorean Identity to Prove Pythagorean Theorem
-1
Extra Example 2: Find Angle Given Cosine and Quadrant
-2
Identity Tan(squared)x+1=Sec(squared)x

23m 16s

Intro
0:00
Main Formulas
0:19
Companion to Pythagorean Identity
0:27
For Cotangents and Cosecants
0:52
How to Remember
0:58
Example 1: Prove the Identity
1:40
Example 2: Given Tan Find Sec
3:42
Example 3: Prove the Identity
7:45
Extra Example 1: Prove the Identity
-1
Extra Example 2: Given Sec Find Tan
-2

52m 52s

Intro
0:00
0:09
How to Remember
0:48
Cofunction Identities
1:31
How to Remember Graphically
1:44
Where to Use Cofunction Identities
2:52
Example 1: Derive the Formula for cos(A-B)
3:08
Example 2: Use Addition and Subtraction Formulas
16:03
Example 3: Use Addition and Subtraction Formulas to Prove Identity
25:11
Extra Example 1: Use cos(A-B) and Cofunction Identities
-1
Extra Example 2: Convert to Radians and use Formulas
-2
Double Angle Formulas

29m 5s

Intro
0:00
Main Formula
0:07
How to Remember from Addition Formula
0:18
Two Other Forms
1:35
Example 1: Find Sine and Cosine of Angle using Double Angle
3:16
Example 2: Prove Trigonometric Identity using Double Angle
9:37
Example 3: Use Addition and Subtraction Formulas
12:38
Extra Example 1: Find Sine and Cosine of Angle using Double Angle
-1
Extra Example 2: Prove Trigonometric Identity using Double Angle
-2
Half-Angle Formulas

43m 55s

Intro
0:00
Main Formulas
0:09
Confusing Part
0:34
Example 1: Find Sine and Cosine of Angle using Half-Angle
0:54
Example 2: Prove Trigonometric Identity using Half-Angle
11:51
Example 3: Prove the Half-Angle Formula for Tangents
18:39
Extra Example 1: Find Sine and Cosine of Angle using Half-Angle
-1
Extra Example 2: Prove Trigonometric Identity using Half-Angle
-2
Section 8: Applications of Trigonometry
Trigonometry in Right Angles

25m 43s

Intro
0:00
Master Formula for Right Angles
0:11
SOHCAHTOA
0:15
Only for Right Triangles
1:26
Example 1: Find All Angles in a Triangle
2:19
Example 2: Find Lengths of All Sides of Triangle
7:39
Example 3: Find All Angles in a Triangle
11:00
Extra Example 1: Find All Angles in a Triangle
-1
Extra Example 2: Find Lengths of All Sides of Triangle
-2
Law of Sines

56m 40s

Intro
0:00
Law of Sines Formula
0:18
SOHCAHTOA
0:27
Any Triangle
0:59
Graphical Representation
1:25
Solving Triangle Completely
2:37
When to Use Law of Sines
2:55
ASA, SAA, SSA, AAA
2:59
SAS, SSS for Law of Cosines
7:11
Example 1: How Many Triangles Satisfy Conditions, Solve Completely
8:44
Example 2: How Many Triangles Satisfy Conditions, Solve Completely
15:30
Example 3: How Many Triangles Satisfy Conditions, Solve Completely
28:32
Extra Example 1: How Many Triangles Satisfy Conditions, Solve Completely
-1
Extra Example 2: How Many Triangles Satisfy Conditions, Solve Completely
-2
Law of Cosines

49m 5s

Intro
0:00
Law of Cosines Formula
0:23
Graphical Representation
0:34
Relates Sides to Angles
1:00
Any Triangle
1:20
Generalization of Pythagorean Theorem
1:32
When to Use Law of Cosines
2:26
SAS, SSS
2:30
Heron's Formula
4:49
Semiperimeter S
5:11
Example 1: How Many Triangles Satisfy Conditions, Solve Completely
5:53
Example 2: How Many Triangles Satisfy Conditions, Solve Completely
15:19
Example 3: Find Area of a Triangle Given All Side Lengths
26:33
Extra Example 1: How Many Triangles Satisfy Conditions, Solve Completely
-1
Extra Example 2: Length of Third Side and Area of Triangle
-2
Finding the Area of a Triangle

27m 37s

Intro
0:00
Master Right Triangle Formula and Law of Cosines
0:19
SOHCAHTOA
0:27
Law of Cosines
1:23
Heron's Formula
2:22
Semiperimeter S
2:37
Example 1: Area of Triangle with Two Sides and One Angle
3:12
Example 2: Area of Triangle with Three Sides
6:11
Example 3: Area of Triangle with Three Sides, No Heron's Formula
8:50
Extra Example 1: Area of Triangle with Two Sides and One Angle
-1
Extra Example 2: Area of Triangle with Two Sides and One Angle
-2
Word Problems and Applications of Trigonometry

34m 25s

Intro
0:00
Formulas to Remember
0:11
SOHCAHTOA
0:15
Law of Sines
0:55
Law of Cosines
1:48
Heron's Formula
2:46
Example 1: Telephone Pole Height
4:01
Example 2: Bridge Length
7:48
Example 3: Area of Triangular Field
14:20
Extra Example 1: Kite Height
-1
Extra Example 2: Roads to a Town
-2
Section 9: Systems of Equations and Inequalities
Systems of Linear Equations

55m 40s

Intro
0:00
Introduction
0:04
Graphs as Location of 'True'
1:49
All Locations that Make the Function True
2:25
Understand the Relationship Between Solutions and the Graph
3:43
Systems as Graphs
4:07
Equations as Lines
4:20
Intersection Point
5:19
Three Possibilities for Solutions
6:17
Independent
6:24
Inconsistent
6:36
Dependent
7:06
Solving by Substitution
8:37
Solve for One Variable
9:07
Substitute into the Second Equation
9:34
Solve for Both Variables
10:12
What If a System is Inconsistent or Dependent?
11:08
No Solutions
11:25
Infinite Solutions
12:30
Solving by Elimination
13:56
Example
14:22
Determining the Number of Solutions
16:30
Why Elimination Makes Sense
17:25
Solving by Graphing Calculator
19:59
Systems with More than Two Variables
23:22
Example 1
25:49
Example 2
30:22
Example 3
34:11
Example 4
38:55
Example 5
46:01
(Non-) Example 6
53:37
Systems of Linear Inequalities

1h 13s

Intro
0:00
Introduction
0:04
Inequality Refresher-Solutions
0:46
Equation Solutions vs. Inequality Solutions
1:02
Essentially a Wide Variety of Answers
1:35
Refresher--Negative Multiplication Flips
1:43
Refresher--Negative Flips: Why?
3:19
Multiplication by a Negative
3:43
The Relationship Flips
3:55
Refresher--Stick to Basic Operations
4:34
Linear Equations in Two Variables
6:50
Graphing Linear Inequalities
8:28
Why It Includes a Whole Section
8:43
How to Show The Difference Between Strict and Not Strict Inequalities
10:08
Dashed Line--Not Solutions
11:10
Solid Line--Are Solutions
11:24
11:42
Example of Using a Point
12:41
13:14
Graphing a System
14:53
Set of Solutions is the Overlap
15:17
Example
15:22
Solutions are Best Found Through Graphing
18:05
Linear Programming-Idea
19:52
Use a Linear Objective Function
20:15
Variables in Objective Function have Constraints
21:24
Linear Programming-Method
22:09
Rearrange Equations
22:21
Graph
22:49
Critical Solution is at the Vertex of the Overlap
23:40
Try Each Vertice
24:35
Example 1
24:58
Example 2
28:57
Example 3
33:48
Example 4
43:10
Nonlinear Systems

41m 1s

Intro
0:00
Introduction
0:06
Substitution
1:12
Example
1:22
Elimination
3:46
Example
3:56
Elimination is Less Useful for Nonlinear Systems
4:56
Graphing
5:56
Using a Graphing Calculator
6:44
Number of Solutions
8:44
Systems of Nonlinear Inequalities
10:02
Graph Each Inequality
10:06
Dashed and/or Solid
10:18
11:14
Example 1
13:24
Example 2
15:50
Example 3
22:02
Example 4
29:06
Example 4, cont.
33:40
Section 10: Vectors and Matrices
Vectors

1h 9m 31s

Intro
0:00
Introduction
0:10
Magnitude of the Force
0:22
Direction of the Force
0:48
Vector
0:52
Idea of a Vector
1:30
How Vectors are Denoted
2:00
Component Form
3:20
Angle Brackets and Parentheses
3:50
Magnitude/Length
4:26
Denoting the Magnitude of a Vector
5:16
Direction/Angle
7:52
Always Draw a Picture
8:50
Component Form from Magnitude & Angle
10:10
Scaling by Scalars
14:06
Unit Vectors
16:26
Combining Vectors - Algebraically
18:10
Combining Vectors - Geometrically
19:54
Resultant Vector
20:46
Alternate Component Form: i, j
21:16
The Zero Vector
23:18
Properties of Vectors
24:20
No Multiplication (Between Vectors)
28:30
Dot Product
29:40
Motion in a Medium
30:10
Fish in an Aquarium Example
31:38
More Than Two Dimensions
33:12
More Than Two Dimensions - Magnitude
34:18
Example 1
35:26
Example 2
38:10
Example 3
45:48
Example 4
50:40
Example 4, cont.
56:07
Example 5
1:01:32
Dot Product & Cross Product

35m 20s

Intro
0:00
Introduction
0:08
Dot Product - Definition
0:42
Dot Product Results in a Scalar, Not a Vector
2:10
Example in Two Dimensions
2:34
Angle and the Dot Product
2:58
The Dot Product of Two Vectors is Deeply Related to the Angle Between the Two Vectors
2:59
Proof of Dot Product Formula
4:14
Won't Directly Help Us Better Understand Vectors
4:18
Dot Product - Geometric Interpretation
4:58
We Can Interpret the Dot Product as a Measure of How Long and How Parallel Two Vectors Are
7:26
Dot Product - Perpendicular Vectors
8:24
If the Dot Product of Two Vectors is 0, We Know They are Perpendicular to Each Other
8:54
Cross Product - Definition
11:08
Cross Product Only Works in Three Dimensions
11:09
Cross Product - A Mnemonic
12:16
The Determinant of a 3 x 3 Matrix and Standard Unit Vectors
12:17
Cross Product - Geometric Interpretations
14:30
The Right-Hand Rule
15:17
Cross Product - Geometric Interpretations Cont.
17:00
Example 1
18:40
Example 2
22:50
Example 3
24:04
Example 4
26:20
Bonus Round
29:18
Proof: Dot Product Formula
29:24
Proof: Dot Product Formula, cont.
30:38
Matrices

54m 7s

Intro
0:00
Introduction
0:08
Definition of a Matrix
3:02
Size or Dimension
3:58
Square Matrix
4:42
Denoted by Capital Letters
4:56
When are Two Matrices Equal?
5:04
Examples of Matrices
6:44
Rows x Columns
6:46
7:48
We Use Capitals to Denote a Matrix and Lower Case to Denotes Its Entries
8:32
Using Entries to Talk About Matrices
10:08
Scalar Multiplication
11:26
Scalar = Real Number
11:34
Example
12:36
13:08
Example
14:22
Matrix Multiplication
15:00
Example
18:52
Matrix Multiplication, cont.
19:58
Matrix Multiplication and Order (Size)
25:26
Make Sure Their Orders are Compatible
25:27
Matrix Multiplication is NOT Commutative
28:20
Example
30:08
Special Matrices - Zero Matrix (0)
32:48
Zero Matrix Has 0 for All of its Entries
32:49
Special Matrices - Identity Matrix (I)
34:14
Identity Matrix is a Square Matrix That Has 1 for All Its Entries on the Main Diagonal and 0 for All Other Entries
34:15
Example 1
36:16
Example 2
40:00
Example 3
44:54
Example 4
50:08
Determinants & Inverses of Matrices

47m 12s

Intro
0:00
Introduction
0:06
Not All Matrices Are Invertible
1:30
What Must a Matrix Have to Be Invertible?
2:08
Determinant
2:32
The Determinant is a Real Number Associated With a Square Matrix
2:38
If the Determinant of a Matrix is Nonzero, the Matrix is Invertible
3:40
Determinant of a 2 x 2 Matrix
4:34
Think in Terms of Diagonals
5:12
Minors and Cofactors - Minors
6:24
Example
6:46
Minors and Cofactors - Cofactors
8:00
Cofactor is Closely Based on the Minor
8:01
Alternating Sign Pattern
9:04
Determinant of Larger Matrices
10:56
Example
13:00
Alternative Method for 3x3 Matrices
16:46
Not Recommended
16:48
Inverse of a 2 x 2 Matrix
19:02
Inverse of Larger Matrices
20:00
Using Inverse Matrices
21:06
When Multiplied Together, They Create the Identity Matrix
21:24
Example 1
23:45
Example 2
27:21
Example 3
32:49
Example 4
36:27
Finding the Inverse of Larger Matrices
41:59
General Inverse Method - Step 1
43:25
General Inverse Method - Step 2
43:27
General Inverse Method - Step 2, cont.
43:27
General Inverse Method - Step 3
45:15
Using Matrices to Solve Systems of Linear Equations

58m 34s

Intro
0:00
Introduction
0:12
Augmented Matrix
1:44
We Can Represent the Entire Linear System With an Augmented Matrix
1:50
Row Operations
3:22
Interchange the Locations of Two Rows
3:50
Multiply (or Divide) a Row by a Nonzero Number
3:58
Add (or Subtract) a Multiple of One Row to Another
4:12
Row Operations - Keep Notes!
5:50
Suggested Symbols
7:08
Gauss-Jordan Elimination - Idea
8:04
Gauss-Jordan Elimination - Idea, cont.
9:16
Reduced Row-Echelon Form
9:18
Gauss-Jordan Elimination - Method
11:36
Begin by Writing the System As An Augmented Matrix
11:38
Gauss-Jordan Elimination - Method, cont.
13:48
Cramer's Rule - 2 x 2 Matrices
17:08
Cramer's Rule - n x n Matrices
19:24
Solving with Inverse Matrices
21:10
Solving Inverse Matrices, cont.
25:28
The Mighty (Graphing) Calculator
26:38
Example 1
29:56
Example 2
33:56
Example 3
37:00
Example 3, cont.
45:04
Example 4
51:28
Section 11: Alternate Ways to Graph
Parametric Equations

53m 33s

Intro
0:00
Introduction
0:06
Definition
1:10
Plane Curve
1:24
The Key Idea
2:00
Graphing with Parametric Equations
2:52
Same Graph, Different Equations
5:04
How Is That Possible?
5:36
Same Graph, Different Equations, cont.
5:42
Here's Another to Consider
7:56
Same Plane Curve, But Still Different
8:10
A Metaphor for Parametric Equations
9:36
Think of Parametric Equations As a Way to Describe the Motion of An Object
9:38
Graph Shows Where It Went, But Not Speed
10:32
Eliminating Parameters
12:14
Rectangular Equation
12:16
Caution
13:52
Creating Parametric Equations
14:30
Interesting Graphs
16:38
Graphing Calculators, Yay!
19:18
Example 1
22:36
Example 2
28:26
Example 3
37:36
Example 4
41:00
Projectile Motion
44:26
Example 5
47:00
Polar Coordinates

48m 7s

Intro
0:00
Introduction
0:04
Polar Coordinates Give Us a Way To Describe the Location of a Point
0:26
Polar Equations and Functions
0:50
Plotting Points with Polar Coordinates
1:06
The Distance of the Point from the Origin
1:09
The Angle of the Point
1:33
Give Points as the Ordered Pair (r,θ)
2:03
Visualizing Plotting in Polar Coordinates
2:32
First Way We Can Plot
2:39
Second Way We Can Plot
2:50
First, We'll Look at Visualizing r, Then θ
3:09
Rotate the Length Counter-Clockwise by θ
3:38
Alternatively, We Can Visualize θ, Then r
4:06
'Polar Graph Paper'
6:17
Horizontal and Vertical Tick Marks Are Not Useful for Polar
6:42
Use Concentric Circles to Helps Up See Distance From the Pole
7:08
Can Use Arc Sectors to See Angles
7:57
Multiple Ways to Name a Point
9:17
Examples
9:30
For Any Angle θ, We Can Make an Equivalent Angle
10:44
Negative Values for r
11:58
If r Is Negative, We Go In The Direction Opposite the One That The Angle θ Points Out
12:22
Another Way to Name the Same Point: Add π to θ and Make r Negative
13:44
Converting Between Rectangular and Polar
14:37
Rectangular Way to Name
14:43
Polar Way to Name
14:52
The Rectangular System Must Have a Right Angle Because It's Based on a Rectangle
15:08
Connect Both Systems Through Basic Trigonometry
15:38
Equation to Convert From Polar to Rectangular Coordinate Systems
16:55
Equation to Convert From Rectangular to Polar Coordinate Systems
17:13
Converting to Rectangular is Easy
17:20
Converting to Polar is a Bit Trickier
17:21
Draw Pictures
18:55
Example 1
19:50
Example 2
25:17
Example 3
31:05
Example 4
35:56
Example 5
41:49
Polar Equations & Functions

38m 16s

Intro
0:00
Introduction
0:04
Equations and Functions
1:16
Independent Variable
1:21
Dependent Variable
1:30
Examples
1:46
Always Assume That θ Is In Radians
2:44
Graphing in Polar Coordinates
3:29
Graph is the Same Way We Graph 'Normal' Stuff
3:32
Example
3:52
Graphing in Polar - Example, Cont.
6:45
Tips for Graphing
9:23
Notice Patterns
10:19
Repetition
13:39
Graphing Equations of One Variable
14:39
Converting Coordinate Types
16:16
Use the Same Conversion Formulas From the Previous Lesson
16:23
Interesting Graphs
17:48
Example 1
18:03
Example 2
18:34
Graphing Calculators, Yay!
19:07
Plot Random Things, Alter Equations You Understand, Get a Sense for How Polar Stuff Works
19:11
Check Out the Appendix
19:26
Example 1
21:36
Example 2
28:13
Example 3
34:24
Example 4
35:52
Section 12: Complex Numbers and Polar Coordinates
Polar Form of Complex Numbers

40m 43s

Intro
0:00
Polar Coordinates
0:49
Rectangular Form
0:52
Polar Form
1:25
R and Theta
1:51
Polar Form Conversion
2:27
R and Theta
2:35
Optimal Values
4:05
Euler's Formula
4:25
Multiplying Two Complex Numbers in Polar Form
6:10
Multiply r's Together and Add Exponents
6:32
Example 1: Convert Rectangular to Polar Form
7:17
Example 2: Convert Polar to Rectangular Form
13:49
Example 3: Multiply Two Complex Numbers
17:28
Extra Example 1: Convert Between Rectangular and Polar Forms
-1
Extra Example 2: Simplify Expression to Polar Form
-2
DeMoivre's Theorem

57m 37s

Intro
0:00
Introduction to DeMoivre's Theorem
0:10
n nth Roots
3:06
DeMoivre's Theorem: Finding nth Roots
3:52
Relation to Unit Circle
6:29
One nth Root for Each Value of k
7:11
Example 1: Convert to Polar Form and Use DeMoivre's Theorem
8:24
Example 2: Find Complex Eighth Roots
15:27
Example 3: Find Complex Roots
27:49
Extra Example 1: Convert to Polar Form and Use DeMoivre's Theorem
-1
Extra Example 2: Find Complex Fourth Roots
-2
Section 13: Counting & Probability
Counting

31m 36s

Intro
0:00
Introduction
0:08
Combinatorics
0:56
Definition: Event
1:24
Example
1:50
Visualizing an Event
3:02
Branching line diagram
3:06
3:40
Example
4:18
Multiplication Principle
5:42
Example
6:24
Pigeonhole Principle
8:06
Example
10:26
Draw Pictures
11:06
Example 1
12:02
Example 2
14:16
Example 3
17:34
Example 4
21:26
Example 5
25:14
Permutations & Combinations

44m 3s

Intro
0:00
Introduction
0:08
Permutation
0:42
Combination
1:10
Towards a Permutation Formula
2:38
How Many Ways Can We Arrange the Letters A, B, C, D, and E?
3:02
Towards a Permutation Formula, cont.
3:34
Factorial Notation
6:56
Symbol Is '!'
6:58
Examples
7:32
Permutation of n Objects
8:44
Permutation of r Objects out of n
9:04
What If We Have More Objects Than We Have Slots to Fit Them Into?
9:46
Permutation of r Objects Out of n, cont.
10:28
Distinguishable Permutations
14:46
What If Not All Of the Objects We're Permuting Are Distinguishable From Each Other?
14:48
Distinguishable Permutations, cont.
17:04
Combinations
19:04
Combinations, cont.
20:56
Example 1
23:10
Example 2
26:16
Example 3
28:28
Example 4
31:52
Example 5
33:58
Example 6
36:34
Probability

36m 58s

Intro
0:00
Introduction
0:06
Definition: Sample Space
1:18
Event = Something Happening
1:20
Sample Space
1:36
Probability of an Event
2:12
Let E Be An Event and S Be The Corresponding Sample Space
2:14
'Equally Likely' Is Important
3:52
Fair and Random
5:26
Interpreting Probability
6:34
How Can We Interpret This Value?
7:24
We Can Represent Probability As a Fraction, a Decimal, Or a Percentage
8:04
One of Multiple Events Occurring
9:52
Mutually Exclusive Events
10:38
What If The Events Are Not Mutually Exclusive?
12:20
Taking the Possibility of Overlap Into Account
13:24
An Event Not Occurring
17:14
Complement of E
17:22
Independent Events
19:36
Independent
19:48
Conditional Events
21:28
What Is The Events Are Not Independent Though?
21:30
Conditional Probability
22:16
Conditional Events, cont.
23:51
Example 1
25:27
Example 2
27:09
Example 3
28:57
Example 4
30:51
Example 5
34:15
Section 14: Conic Sections
Parabolas

41m 27s

Intro
0:00
What is a Parabola?
0:20
Definition of a Parabola
0:29
Focus
0:59
Directrix
1:15
Axis of Symmetry
3:08
Vertex
3:33
Minimum or Maximum
3:44
Standard Form
4:59
Horizontal Parabolas
5:08
Vertex Form
5:19
Upward or Downward
5:41
Example: Standard Form
6:06
Graphing Parabolas
8:31
Shifting
8:51
Example: Completing the Square
9:22
Symmetry and Translation
12:18
Example: Graph Parabola
12:40
Latus Rectum
17:13
Length
18:15
Example: Latus Rectum
18:35
Horizontal Parabolas
18:57
Not Functions
20:08
Example: Horizontal Parabola
21:21
Focus and Directrix
24:11
Horizontal
24:48
Example 1: Parabola Standard Form
25:12
Example 2: Graph Parabola
30:00
Example 3: Graph Parabola
33:13
Example 4: Parabola Equation
37:28
Circles

21m 3s

Intro
0:00
What are Circles?
0:08
Example: Equidistant
0:17
0:32
Equation of a Circle
0:44
Example: Standard Form
1:11
Graphing Circles
1:47
Example: Circle
1:56
Center Not at Origin
3:07
Example: Completing the Square
3:51
Example 1: Equation of Circle
6:44
11:51
15:08
Example 4: Equation of Circle
16:57
Ellipses

46m 51s

Intro
0:00
What Are Ellipses?
0:11
Foci
0:23
Properties of Ellipses
1:43
Major Axis, Minor Axis
1:47
Center
1:54
Length of Major Axis and Minor Axis
3:21
Standard Form
5:33
Example: Standard Form of Ellipse
6:09
Vertical Major Axis
9:14
Example: Vertical Major Axis
9:46
Graphing Ellipses
12:51
Complete the Square and Symmetry
13:00
Example: Graphing Ellipse
13:16
Equation with Center at (h, k)
19:57
Horizontal and Vertical
20:14
Difference
20:27
Example: Center at (h, k)
20:55
Example 1: Equation of Ellipse
24:05
Example 2: Equation of Ellipse
27:57
Example 3: Equation of Ellipse
32:32
Example 4: Graph Ellipse
38:27
Hyperbolas

38m 15s

Intro
0:00
What are Hyperbolas?
0:12
Two Branches
0:18
Foci
0:38
Properties
2:00
Transverse Axis and Conjugate Axis
2:06
Vertices
2:46
Length of Transverse Axis
3:14
Distance Between Foci
3:31
Length of Conjugate Axis
3:38
Standard Form
5:45
Vertex Location
6:36
Known Points
6:52
Vertical Transverse Axis
7:26
Vertex Location
7:50
Asymptotes
8:36
Vertex Location
8:56
Rectangle
9:28
Diagonals
10:29
Graphing Hyperbolas
12:58
Example: Hyperbola
13:16
Equation with Center at (h, k)
16:32
Example: Center at (h, k)
17:21
Example 1: Equation of Hyperbola
19:20
Example 2: Equation of Hyperbola
22:48
Example 3: Graph Hyperbola
26:05
Example 4: Equation of Hyperbola
36:29
Conic Sections

18m 43s

Intro
0:00
Conic Sections
0:16
Double Cone Sections
0:24
Standard Form
1:27
General Form
1:37
Identify Conic Sections
2:16
B = 0
2:50
X and Y
3:22
Identify Conic Sections, Cont.
4:46
Parabola
5:17
Circle
5:51
Ellipse
6:31
Hyperbola
7:10
Example 1: Identify Conic Section
8:01
Example 2: Identify Conic Section
11:03
Example 3: Identify Conic Section
11:38
Example 4: Identify Conic Section
14:50
Section 15: Sequences, Series, & Induction
Introduction to Sequences

57m 45s

Intro
0:00
Introduction
0:06
Definition: Sequence
0:28
Infinite Sequence
2:08
Finite Sequence
2:22
Length
2:58
Formula for the nth Term
3:22
Defining a Sequence Recursively
5:54
Initial Term
7:58
Sequences and Patterns
10:40
First, Identify a Pattern
12:52
How to Get From One Term to the Next
17:38
Tips for Finding Patterns
19:52
More Tips for Finding Patterns
24:14
Even More Tips
26:50
Example 1
30:32
Example 2
34:54
Fibonacci Sequence
34:55
Example 3
38:40
Example 4
45:02
Example 5
49:26
Example 6
51:54
Introduction to Series

40m 27s

Intro
0:00
Introduction
0:06
Definition: Series
1:20
Why We Need Notation
2:48
Simga Notation (AKA Summation Notation)
4:44
Thing Being Summed
5:42
Index of Summation
6:21
Lower Limit of Summation
7:09
Upper Limit of Summation
7:23
Sigma Notation, Example
7:36
Sigma Notation for Infinite Series
9:08
How to Reindex
10:58
How to Reindex, Expanding
12:56
How to Reindex, Substitution
16:46
Properties of Sums
19:42
Example 1
23:46
Example 2
25:34
Example 3
27:12
Example 4
29:54
Example 5
32:06
Example 6
37:16
Arithmetic Sequences & Series

31m 36s

Intro
0:00
Introduction
0:05
Definition: Arithmetic Sequence
0:47
Common Difference
1:13
Two Examples
1:19
Form for the nth Term
2:14
Recursive Relation
2:33
Towards an Arithmetic Series Formula
5:12
Creating a General Formula
10:09
General Formula for Arithmetic Series
14:23
Example 1
15:46
Example 2
17:37
Example 3
22:21
Example 4
24:09
Example 5
27:14
Geometric Sequences & Series

39m 27s

Intro
0:00
Introduction
0:06
Definition
0:48
Form for the nth Term
2:42
Formula for Geometric Series
5:16
Infinite Geometric Series
11:48
Diverges
13:04
Converges
14:48
Formula for Infinite Geometric Series
16:32
Example 1
20:32
Example 2
22:02
Example 3
26:00
Example 4
30:48
Example 5
34:28
Mathematical Induction

49m 53s

Intro
0:00
Introduction
0:06
Belief Vs. Proof
1:22
A Metaphor for Induction
6:14
The Principle of Mathematical Induction
11:38
Base Case
13:24
Inductive Step
13:30
Inductive Hypothesis
13:52
A Remark on Statements
14:18
Using Mathematical Induction
16:58
Working Example
19:58
Finding Patterns
28:46
Example 1
30:17
Example 2
37:50
Example 3
42:38
The Binomial Theorem

1h 13m 13s

Intro
0:00
Introduction
0:06
We've Learned That a Binomial Is An Expression That Has Two Terms
0:07
Understanding Binomial Coefficients
1:20
Things We Notice
2:24
What Goes In the Blanks?
5:52
Each Blank is Called a Binomial Coefficient
6:18
The Binomial Theorem
6:38
Example
8:10
The Binomial Theorem, cont.
10:46
We Can Also Write This Expression Compactly Using Sigma Notation
12:06
Proof of the Binomial Theorem
13:22
Proving the Binomial Theorem Is Within Our Reach
13:24
Pascal's Triangle
15:12
Pascal's Triangle, cont.
16:12
16:24
Zeroth Row
18:04
First Row
18:12
Why Do We Care About Pascal's Triangle?
18:50
Pascal's Triangle, Example
19:26
Example 1
21:26
Example 2
24:34
Example 3
28:34
Example 4
32:28
Example 5
37:12
Time for the Fireworks!
43:38
Proof of the Binomial Theorem
43:44
We'll Prove This By Induction
44:04
Proof (By Induction)
46:36
Proof, Base Case
47:00
Proof, Inductive Step - Notation Discussion
49:22
Induction Step
49:24
Proof, Inductive Step - Setting Up
52:26
Induction Hypothesis
52:34
What We What To Show
52:44
Proof, Inductive Step - Start
54:18
Proof, Inductive Step - Middle
55:38
Expand Sigma Notations
55:48
Proof, Inductive Step - Middle, cont.
58:40
Proof, Inductive Step - Checking In
1:01:08
Let's Check In With Our Original Goal
1:01:12
Want to Show
1:01:18
Lemma - A Mini Theorem
1:02:18
Proof, Inductive Step - Lemma
1:02:52
Proof of Lemma: Let's Investigate the Left Side
1:03:08
Proof, Inductive Step - Nearly There
1:07:54
Proof, Inductive Step - End!
1:09:18
Proof, Inductive Step - End!, cont.
1:11:01
Section 16: Preview of Calculus
Idea of a Limit

40m 22s

Intro
0:00
Introduction
0:05
Motivating Example
1:26
Fuzzy Notion of a Limit
3:38
Limit is the Vertical Location a Function is Headed Towards
3:44
Limit is What the Function Output is Going to Be
4:15
Limit Notation
4:33
Exploring Limits - 'Ordinary' Function
5:26
Test Out
5:27
Graphing, We See The Answer Is What We Would Expect
5:44
Exploring Limits - Piecewise Function
6:45
If We Modify the Function a Bit
6:49
Exploring Limits - A Visual Conception
10:08
Definition of a Limit
12:07
If f(x) Becomes Arbitrarily Close to Some Number L as x Approaches Some Number c, Then the Limit of f(x) As a Approaches c is L.
12:09
We Are Not Concerned with f(x) at x=c
12:49
We Are Considering x Approaching From All Directions, Not Just One Side
13:10
Limits Do Not Always Exist
15:47
Finding Limits
19:49
Graphs
19:52
Tables
21:48
Precise Methods
24:53
Example 1
26:06
Example 2
27:39
Example 3
30:51
Example 4
33:11
Example 5
37:07
Formal Definition of a Limit

57m 11s

Intro
0:00
Introduction
0:06
New Greek Letters
2:42
Delta
3:14
Epsilon
3:46
Sometimes Called the Epsilon-Delta Definition of a Limit
3:56
Formal Definition of a Limit
4:22
What does it MEAN!?!?
5:00
The Groundwork
5:38
Set Up the Limit
5:39
The Function is Defined Over Some Portion of the Reals
5:58
The Horizontal Location is the Value the Limit Will Approach
6:28
The Vertical Location L is Where the Limit Goes To
7:00
The Epsilon-Delta Part
7:26
The Hard Part is the Second Part of the Definition
7:30
Second Half of Definition
10:04
Restrictions on the Allowed x Values
10:28
The Epsilon-Delta Part, cont.
13:34
Sherlock Holmes and Dr. Watson
15:08
The Adventure of the Delta-Epsilon Limit
15:16
Setting
15:18
We Begin By Setting Up the Game As Follows
15:52
The Adventure of the Delta-Epsilon, cont.
17:24
17:46
What If I Try Larger?
19:39
Technically, You Haven't Proven the Limit
20:53
Here is the Method
21:18
What We Should Concern Ourselves With
22:20
Investigate the Left Sides of the Expressions
25:24
We Can Create the Following Inequalities
28:08
Finally…
28:50
Nothing Like a Good Proof to Develop the Appetite
30:42
Example 1
31:02
Example 1, cont.
36:26
Example 2
41:46
Example 2, cont.
47:50
Finding Limits

32m 40s

Intro
0:00
Introduction
0:08
Method - 'Normal' Functions
2:04
The Easiest Limits to Find
2:06
It Does Not 'Break'
2:18
It Is Not Piecewise
2:26
Method - 'Normal' Functions, Example
3:38
Method - 'Normal' Functions, cont.
4:54
The Functions We're Used to Working With Go Where We Expect Them To Go
5:22
5:42
Method - Canceling Factors
7:18
One Weird Thing That Often Happens is Dividing By 0
7:26
Method - Canceling Factors, cont.
8:16
Notice That The Two Functions Are Identical With the Exception of x=0
8:20
Method - Canceling Factors, cont.
10:00
Example
10:52
Method - Rationalization
12:04
Rationalizing a Portion of Some Fraction
12:05
Conjugate
12:26
Method - Rationalization, cont.
13:14
Example
13:50
Method - Piecewise
16:28
The Limits of Piecewise Functions
16:30
Example 1
17:42
Example 2
18:44
Example 3
20:20
Example 4
22:24
Example 5
24:24
Example 6
27:12
Continuity & One-Sided Limits

32m 43s

Intro
0:00
Introduction
0:06
Motivating Example
0:56
Continuity - Idea
2:14
Continuous Function
2:18
All Parts of Function Are Connected
2:28
Function's Graph Can Be Drawn Without Lifting Pencil
2:36
There Are No Breaks or Holes in Graph
2:56
Continuity - Idea, cont.
3:38
We Can Interpret the Break in the Continuity of f(x) as an Issue With the Function 'Jumping'
3:52
Continuity - Definition
5:16
A Break in Continuity is Caused By the Limit Not Matching Up With What the Function Does
5:18
Discontinuous
6:02
Discontinuity
6:10
Continuity and 'Normal' Functions
6:48
Return of the Motivating Example
8:14
One-Sided Limit
8:48
One-Sided Limit - Definition
9:16
Only Considers One Side
9:20
Be Careful to Keep Track of Which Symbol Goes With Which Side
10:06
One-Sided Limit - Example
10:50
There Does Not Necessarily Need to Be a Connection Between Left or Right Side Limits
11:16
Normal Limits and One-Sided Limits
12:08
Limits of Piecewise Functions
14:12
'Breakover' Points
14:22
We Find the Limit of a Piecewise Function By Checking If the Left and Right Side Limits Agree With Each Other
15:34
Example 1
16:40
Example 2
18:54
Example 3
22:00
Example 4
26:36
Limits at Infinity & Limits of Sequences

32m 49s

Intro
0:00
Introduction
0:06
Definition: Limit of a Function at Infinity
1:44
A Limit at Infinity Works Very Similarly to How a Normal Limit Works
2:38
Evaluating Limits at Infinity
4:08
Rational Functions
4:17
Examples
4:30
For a Rational Function, the Question Boils Down to Comparing the Long Term Growth Rates of the Numerator and Denominator
5:22
There are Three Possibilities
6:36
Evaluating Limits at Infinity, cont.
8:08
Does the Function Grow Without Bound? Will It 'Settle Down' Over Time?
10:06
10:26
Limit of a Sequence
12:20
What Value Does the Sequence Tend to Do in the Long-Run?
12:41
The Limit of a Sequence is Very Similar to the Limit of a Function at Infinity
12:52
Numerical Evaluation
14:16
Numerically: Plug in Numbers and See What Comes Out
14:24
Example 1
16:42
Example 2
21:00
Example 3
22:08
Example 4
26:14
Example 5
28:10
Example 6
31:06
Instantaneous Slope & Tangents (Derivatives)

51m 13s

Intro
0:00
Introduction
0:08
The Derivative of a Function Gives Us a Way to Talk About 'How Fast' the Function If Changing
0:16
Instantaneous Slop
0:22
Instantaneous Rate of Change
0:28
Slope
1:24
The Vertical Change Divided by the Horizontal
1:40
Idea of Instantaneous Slope
2:10
What If We Wanted to Apply the Idea of Slope to a Non-Line?
2:14
Tangent to a Circle
3:52
What is the Tangent Line for a Circle?
4:42
Tangent to a Curve
5:20
Towards a Derivative - Average Slope
6:36
Towards a Derivative - Average Slope, cont.
8:20
An Approximation
11:24
Towards a Derivative - General Form
13:18
Towards a Derivative - General Form, cont.
16:46
An h Grows Smaller, Our Slope Approximation Becomes Better
18:44
Towards a Derivative - Limits!
20:04
Towards a Derivative - Limits!, cont.
22:08
We Want to Show the Slope at x=1
22:34
Towards a Derivative - Checking Our Slope
23:12
Definition of the Derivative
23:54
Derivative: A Way to Find the Instantaneous Slope of a Function at Any Point
23:58
Differentiation
24:54
Notation for the Derivative
25:58
The Derivative is a Very Important Idea In Calculus
26:04
The Important Idea
27:34
Why Did We Learn the Formal Definition to Find a Derivative?
28:18
Example 1
30:50
Example 2
36:06
Example 3
40:24
The Power Rule
44:16
Makes It Easier to Find the Derivative of a Function
44:24
Examples
45:04
n Is Any Constant Number
45:46
Example 4
46:26
Area Under a Curve (Integrals)

45m 26s

Intro
0:00
Introduction
0:06
Integral
0:12
Idea of Area Under a Curve
1:18
Approximation by Rectangles
2:12
The Easiest Way to Find Area is With a Rectangle
2:18
Various Methods for Choosing Rectangles
4:30
Rectangle Method - Left-Most Point
5:12
The Left-Most Point
5:16
Rectangle Method - Right-Most Point
5:58
The Right-Most Point
6:00
Rectangle Method - Mid-Point
6:42
Horizontal Mid-Point
6:48
Rectangle Method - Maximum (Upper Sum)
7:34
Maximum Height
7:40
Rectangle Method - Minimum
8:54
Minimum Height
9:02
Evaluating the Area Approximation
10:08
Split the Interval Into n Sub-Intervals
10:30
More Rectangles, Better Approximation
12:14
The More We Us , the Better Our Approximation Becomes
12:16
Our Approximation Becomes More Accurate as the Number of Rectangles n Goes Off to Infinity
12:44
Finding Area with a Limit
13:08
If This Limit Exists, It Is Called the Integral From a to b
14:08
The Process of Finding Integrals is Called Integration
14:22
The Big Reveal
14:40
The Integral is Based on the Antiderivative
14:46
The Big Reveal - Wait, Why?
16:28
The Rate of Change for the Area is Based on the Height of the Function
16:50
Height is the Derivative of Area, So Area is Based on the Antiderivative of Height
17:50
Example 1
19:06
Example 2
22:48
Example 3
29:06
Example 3, cont.
35:14
Example 4
40:14
Section 17: Appendix: Graphing Calculators

10m 41s

Intro
0:00
0:06
Should I Get a Graphing Utility?
0:20
Free Graphing Utilities - Web Based
0:38
Personal Favorite: Desmos
0:58
Free Graphing Utilities - Offline Programs
1:18
GeoGebra
1:31
Microsoft Mathematics
1:50
Grapher
2:18
Other Graphing Utilities - Tablet/Phone
2:48
Should You Buy a Graphing Calculator?
3:22
The Only Real Downside
4:10
4:20
If You Plan on Continuing in Math and/or Science
4:26
If Money is Not Particularly Tight for You
4:32
If You Don't Plan to Continue in Math and Science
5:02
If You Do Plan to Continue and Money Is Tight
5:28
5:44
Which Graphing Calculator is Best?
5:46
Too Many Factors
5:54
6:12
The Old Standby
7:10
TI-83 (Plus)
7:16
TI-84 (Plus)
7:18
9:17
9:19
9:35
10:09
Graphing Calculator Basics

10m 51s

Intro
0:00
0:06
Skim It
0:20
Play Around and Experiment
0:34
Syntax
0:40
Definition of Syntax in English and Math
0:46
Pay Careful Attention to Your Syntax When Working With a Calculator
2:08
Make Sure You Use Parentheses to Indicate the Proper Order of Operations
2:16
3:54
Settings
4:58
You'll Almost Never Need to Change the Settings on Your Calculator
5:00
Tell Calculator In Settings Whether the Angles Are In Radians or Degrees
5:26
Graphing Mode
6:32
Error Messages
7:10
Don't Panic
7:11
Internet Search
7:32
So Many Things
8:14
More Powerful Than You Realize
8:18
Other Things Your Graphing Calculator Can Do
8:24
Playing Around
9:16
Graphing Functions, Window Settings, & Table of Values

10m 38s

Intro
0:00
Graphing Functions
0:18
Graphing Calculator Expects the Variable to Be x
0:28
Syntax
0:58
The Syntax We Choose Will Affect How the Function Graphs
1:00
Use Parentheses
1:26
The Viewing Window
2:00
One of the Most Important Ideas When Graphing Is To Think About The Viewing Window
2:01
For Example
2:30
The Viewing Window, cont.
2:36
Window Settings
3:24
Manually Choose Window Settings
4:20
x Min
4:40
x Max
4:42
y Min
4:44
y Max
4:46
Changing the x Scale or y Scale
5:08
Window Settings, cont.
5:44
Table of Values
7:38
Allows You to Quickly Churn Out Values for Various Inputs
7:42
For example
7:44
Changing the Independent Variable From 'Automatic' to 'Ask'
8:50
Finding Points of Interest

9m 45s

Intro
0:00
Points of Interest
0:06
Interesting Points on the Graph
0:11
Roots/Zeros (Zero)
0:18
Relative Minimums (Min)
0:26
Relative Maximums (Max)
0:32
Intersections (Intersection)
0:38
Finding Points of Interest - Process
1:48
Graph the Function
1:49
2:12
Choose Point of Interest Type
2:54
Identify Where Search Should Occur
3:04
Give a Guess
3:36
Get Result
4:06
5:10
Find Out What Input Value Causes a Certain Output
5:12
For Example
5:24
7:18
Derivative
7:22
Integral
7:30
But How Do You Show Work?
8:20
Parametric & Polar Graphs

7m 8s

Intro
0:00
Change Graph Type
0:08
Located in General 'Settings'
0:16
Graphing in Parametric
1:06
Set Up Both Horizontal Function and Vertical Function
1:08
For Example
2:04
Graphing in Polar
4:00
For Example
4:28

• ## Related Books

### Roots (Zeros) of Polynomials

• The roots/zeros/x-intercepts of a polynomial are the x-values where the polynomial equals 0.
• If you have access to the graph of a function/equation, it is very easy to see where the roots are: where the graph cuts the horizontal axis (the x-intercepts)! Why? Because that's where f(x) = 0 or y=0.
• While you can occasionally find the roots to a polynomial by trying to isolate the variable and directly solve for it, that method often fails or is misleading.
• To find the roots of a polynomial we need to factor the polynomial: break it into its multiplicative factors. Then we can set each factor to 0 and solve to find the roots.
• Factoring can be quite difficult if you're trying to factor a very large or complicated polynomial. There is no procedure that will work for factoring all polynomials.
• In general, if we have a quadratic trinomial (something in the form ax2 + bx + c), we can factor it into a pair of linear binomials as
 ax2 + bx + c  = ( x +  ) ( x +  ).
Think about what has to go in each blank for it to be equivalent to the polynomial you started with.
• Whenever you're factoring polynomials, make sure you check your work! Even on an easy problem, there are ample opportunities to make a mistake. That means you should always try expanding the polynomial (it's fine to do it in your head) to make sure you factored it correctly.
• In general, factoring higher degree polynomials is similar to what we did above. Figure out how you can break down the polynomial into a structure like the above, then ask yourself how you can fill in the blanks.
• If you already know a root to a polynomial, it must be one of its factors. For example, if we know x=a is a root, then the polynomial must have a factor of (x−a). This makes factoring the polynomial that much easier.
• Not all polynomials can be factored. Sometimes it is impossible to reduce it to smaller factors. In such a case, we call the polynomial irreducible. [Later on, we'll discuss a a hidden type of number we haven't previously explored when we learn about the complex numbers. These will allow us to factor these supposedly irreducible polynomials. However, for the most part, we won't work with complex numbers so such polynomials will stay irreducible.]
• There is a limit to how many roots/factors a polynomial can have. A polynomial of degree n can have, at most, n roots/factors.
• We also get information about the possible shape of a polynomial's graph from its degree. A polynomial of degree n can have, at most, n−1 peaks and valleys (formally speaking, relative maximums and minimums).

### Roots (Zeros) of Polynomials

Find the zeros of f(x) = x2+2x − 15.
• The zeros of f(x) are all those x where f(x)=0. That means we want to find all the x such that
 0 = x2+2x − 15.
• We do this by factoring the polynomial. Break the polynomial into factors. Because we have a quadratic, we know it will probably break into factors of the form
 ( x +  ) ( x +  ) .
• We know that the coefficients in the front blanks for each parenthetical must multiply to make 1, because that is the coefficient in front of x2. We know the numbers in the other blanks must multiply to make −15. Finally, we know that when we add those numbers together, they must make 2 because we have 2x.
• Working through this, we find (x−3)(x+5) works, which we can check. If we expand those factors, they do indeed come out to be x2 + 2x −15, so we know the factoring is correct.
• Once we have 0=(x−3)(x+5), we can find the values for x. We set each of the factors equal to 0. [We can do this by the same logic that 0 = a·b means a and/or b must be 0.]
 0 = x−3        0 = x+5
x = −5,  3
Find the roots of g(t) = t2 − 7t − 18.
• The roots of g(t) mean the same thing as the zeros: all those t where g(t)=0. That means we want to find all the t such that
 0 = t2 − 7t − 18.
• We do this by factoring the polynomial. Break the polynomial into factors. Because we have a quadratic, we know it will probably break into factors of the form
 ( t +  ) ( t +  ) .
• We find (t+2)(t−9) works, which we can check. If we expand those factors, they do indeed come out to be t2 − 7t − 18, so we know the factoring is correct.
• Once we have 0=(t+2)(t−9), we can find the values for t. We set each of the factors equal to 0. [We can do this by the same logic that 0 = a·b means a and/or b must be 0.]
 0 = t+2        0 = t−9
t = −2,  9
Solve for x:    2x2 + 11x + 30 = x2 −4x −20.
• Because we're trying to solve something involving polynomials, we will need to use factoring. However, before factoring can be useful, we need to get one side equal to 0. [This is because we want to eventually have something of the form 0 = a·b and then say 0=a and 0=b.]
• Use algebra to get one side of the equation equal to 0. We obtain
 x2 + 15x + 50 = 0.
• Factor the left side to get
 (x+10)(x+5) = 0.
• Now that we have it in the form of 0 equals things multiplied together, we can set each of those things equal to 0:
 0 = x+10        0 = x+5
x = −10, −5
Find the zeros of h(x) = x3 − 9x2 + 23x − 15.
• The zeros of h(x) are all those x where h(x)=0. That means we want to find all the x such that
 0 = x3 − 9x2 + 23x − 15.
• We do this by factoring the polynomial. Break the polynomial into factors. Start off easy by trying to break it into something of the form
 ( x +  )  ( x2 +  x +  ).
• While there are other possibilities (depending on which factor we pull out first), we might get
 (x−1)(x2−8x+15).
[With something difficult like this, it's even more important to check your work and expand the polynomial in your head to be sure you didn't make a mistake.]
• Now factor the remaining x2−8x+15 to obtain
 (x−1)(x−3)(x−5).
• We now have 0 = (x−1)(x−3)(x−5). Set each of the three factors to 0 and solve:
 0 = x−1        0 = x−3        0 = x−5
x=1,  3,  5
Find the other roots to y = x3 + 4x2 − 7x −10 if one of the roots is x=2.
• If you know a root, that automatically implies a factor. A root of x=a implies a factor of (x−a). Thus, since we know x=2 is a root of the polynomial, we know (x−2) is a factor, which will help us in our factoring.
• Pulling out (x−2), we know we'll have something of the form
 (x−2)  ( x2 +  x +  ),
now we just need to appropriately fill in the blanks.
• Begin by filling in the blanks that are easiest. You know the coefficient on x2 must be 1 because in the fully expanded polynomial it is x3. Furthermore, you know the blank for the constant must be 5 because in the fully expanded polynomial you have −10. Then use that information to figure out the middle blank. Eventually you obtain
 (x−2)(x2 + 6x +5).
• Factor the right parenthetical to finish factoring the entire polynomial:
 (x−2)(x+1)(x+5)
• Thus, to find the roots, we have 0 = (x−2)(x+1)(x+5). Set each of these equal to 0, then solve:
 0 = x−2       0 = x+1        0 = x+5
The additional roots are x=−5, −1 (and there is still the root that was given to us in the problem: x=2).
Find all the roots of f(p) = 2 p4 − 8p3 − 14p2 + 44p + 48 given that f(3) = 0 and f(−2) = 0.
• If you know a root, you automatically know a factor. Remember, a root/zero is where the function equals 0. Thus, since we know f(3) = 0 and f(−2) = 0, we know p=3 and p=−2 are roots. A root of p=a implies a factor of (p−a). Therefore we already know two roots: (p−3) and (p+2).
• If we pull out those two factors, we'll have something of the form
 (p−3)(p+2) ( p2 +  p +  ).
• As it stands, it's a little difficult to fill in the blanks. To make it easier to see how the blanks should be filled in, expand the left side:
 (p2 − p − 6)  ( p2 +  p+  )
• Working it out, we obtain
 (p2 − p − 6)  (2p2−6p−8).
• Finish factoring the above:
 (p−3)(p+2)(2p−8)(p+1)
[There are actually other ways to factor the right side. You could also obtain
 (p−3)(p+2)(p−4)(2p+2)     or     2·(p−3)(p+2)(p−4)(p+1)
Still, whatever way you factor it, you will wind up getting the same answer for the roots.]
• We now are looking to solve 0 = (p−3)(p+2)(2p−8)(p+1). Set each factor to 0:
 0 = p−3        0 = p+2        0 = 2p−8        0 = p+1
p=−2, −1, 3,  4
Give a polynomial that has degree 3 and roots at x=−4, 3,  9.
• Each root implies a factor where the constant has the opposite sign:
 x=−4, 3,  9     ⇒     (x+4),    (x−3),    (x−9)
• If we multiply them all together, we have a polynomial with degree 3:
 (x+4)(x−3)(x−9)
• OPTIONAL: You could expand the polynomial if you wanted (or if a problem required it).
• OPTIONAL: If you wanted, you could multiply the entire polynomial by a constant number because it has no effect on the roots. Notice that a·(x+4)(x−3)(x−9) has the exact same roots, because the same values for x will cause the expression to become 0.
(x+4)(x−3)(x−9) or, equivalently, x3−8x2−21x+108
[Additionally, it would also be correct to have a constant multiple of the above, such as −2·(x+4)(x−3)(x−9) = −2x3+16x2 +42x −216.]
Give a polynomial that has degree 4 and roots at x=2, 5.
• Each root implies a factor where the constant has the opposite sign:
 x=2,  5     ⇒     (x−2),    (x−5)
• However, if we just multiply them all together, we only have a polynomial with degree 2:
 (x−2)(x−5)
• We need to increase the degree of the polynomial, but without adding any additional roots. We can do this by adding multiplicity to a single root. Have the same root show up multiple times in the expansion, such as
 (x−2)(x−5)(x−5)(x−5).
In such a way, we now have a polynomial of degree 4.
• OPTIONAL: You could expand the polynomial if you wanted (or if a problem required it).
• OPTIONAL: If you wanted, you could multiply the entire polynomial by a constant number because it has no effect on the roots.
• OPTIONAL: Instead of just using multiplicity and having roots appear more than once, you could also use a polynomial factor that is irreducible, such as (x2 +1). Because we cannot solve 0=x2+1 in the real numbers, it will not add any additional roots. For example, we could have a polynomial like
 (x−2)(x−5)(x2+1)
There are multiple possible answers: any polynomial that contains factors of (x−2) and (x−5) (but no other factors [unless irreducible]) where the total number of factors is 4. Some examples:
 (x−2)3 (x−5)           (x−2)2 (x−5)2
[There are some other optional ways to answer this problem. See the steps above if you're curious.]
What is the maximum number of roots the below polynomial can have?
 x5 − 8x4 + 2x3 + 34x2 + x + 42
• The maximum number of roots/factors a polynomial can have is based on its degree. A polynomial of degree n can have, at most, n roots/factors. Notice that this is a maximum: there is no guarantee it will have that many, we can only be sure it will not have more than that.
• The degree of the polynomial in this problem is 5, so the maximum number of roots is 5.
• In actuality, if you factor this polynomial, it does not have 5 roots. It only has 3 roots because it factors to
 (x2+1)(x−3)(x−7)(x+2).
While (x−3), (x−7), and (x+2) all produce roots [x=3, 7, −2], the factor (x2+1) does not produce any. Why? Because x2+1=0 cannot be solved with any real numbers, so it is irreducible. We will learn more about this idea in later lessons.
The polynomial can not have more than 5 roots.
[If you're curious about the precise number of roots that it has, check out the final step for this problem.]
The area of a rectangle is 66 m2. If the length of the rectangle is 5 m longer than the width, what are the dimensions of the rectangle?
• Begin by setting up the varaibles:
 w = width       l = length
• Create equations from the information given:
 w ·l = 66        w+5 = l
• Notice that we can plug in w+5 for l in the area equation to produce:
 w (w+5) = 66
• This is a quadratic equation, so we can solve it by factoring. Begin by getting 0 on one side of the equation, then factor it:
 w (w+5) = 66     ⇒     w2 +5w − 66=0     ⇒     (w+11)(w−6) = 0
• We have two possibilities for w: w = −11 and w=6. Notice that a negative value for length makes no sense, so we throw out that extraneous solution. This leaves us with w=6.
• Once we know the width, we can find the length. Don't forget to put units in your answer.
The width is 6 m and the length is 11  m.

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

### Roots (Zeros) of Polynomials

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

• Intro 0:00
• Introduction 0:05
• Roots in Graphs 1:17
• The x-intercepts
• How to Remember What 'Roots' Are
• Naïve Attempts 2:31
• Isolating Variables
• Failures of Isolating Variables
• Missing Solutions
• Factoring: How to Find Roots 6:28
• How Factoring Works
• Why Factoring Works
• Steps to Finding Polynomial Roots
• Factoring: How to Find Roots CAUTION 10:08
• Factoring is Not Easy 11:32
• Form of Factored Binomials
• Factoring Examples
• Factoring Higher Degree Polynomials 18:19
• Factoring a Cubic
• Factoring: Roots Imply Factors 19:54
• Where a Root is, A Factor Is
• How to Use Known Roots to Make Factoring Easier
• Not all Polynomials Can be Factored 22:30
• Irreducible Polynomials
• Complex Numbers Help
• Max Number of Roots/Factors 24:57
• Limit to Number of Roots Equal to the Degree
• Why there is a Limit
• Max Number of Peaks/Valleys 26:39
• Shape Information from Degree
• Example Graph
• Max, But Not Required 28:00
• Example 1 28:37
• Example 2 31:21
• Example 3 36:12
• Example 4 38:40

### Transcription: Roots (Zeros) of Polynomials

Welcome back to Educator.com.0000

Today, we are going to talk about roots (also called zeroes) of polynomials.0002

We briefly went over what a root is when we talked about the properties of functions.0006

But let's remind ourselves: the zeroes of a function, the roots of an equation, and the x-intercepts of a graph are all the same thing.0009

They are inputs (which are just x-values) where the output is 0; it is where it comes out to be 0--things that will make our expression give out 0.0017

The roots, zeroes, x-intercepts of f(x) = x2 - 1 and y = x2 - 1 are x = -1 and x = +1,0026

because those two values, f(x) and y, are equal to 0.0036

If we plug in -1, (-1)2 will become positive 1, minus 1 is 0.0040

If we plug in positive 1, 12 is 1, minus 1 is 0.0045

Those two things cause it to give out 0, and the roots of the polynomial are the x-values where the polynomial equals 0.0049

The roots of the polynomial x2 - 1 are -1 and positive 1.0055

Being able to find roots in functions is important for many reasons; and it will come up very often when you are working with polynomials.0062

If you continue on to calculus, you will see how roots can be useful for finding lots of information about a function.0068

So, it is very important to have a grasp of what is going on there and be able to find roots.0073

Roots in graphs: if you have access to the graph of a function/equation, it is very easy to see where the roots are.0078

Of course, you might not see precisely, because it is a graph, after all, and it might be off by a little bit.0083

But you can get a very good sense of where they are: it is where the graph cuts the horizontal axis--the x-intercepts.0088

Why? Because here we have f(x) = 0 or y = 0, depending on if it is a function or an equation.0094

Since that means our height is at 0 there, then every place where we cross the x-axis must be a root--it's as simple as that.0101

This also gives us a nice mnemonic to remember what the word "root" means--0111

it can be a little hard to remember the word "root," since we aren't that used to using it.0115

But we could remember it as where the equation or the function grows out of the x-axis--where it is 0.0118

It is like it is the ground; think of it as a plant rooted in the ground.0125

A function or equation has its roots in the x-axis; a tree has its roots in the earth, and a function has its roots in a height of 0.0129

So, a root is where it is growing up and down; it is where it is held in our plane, held in our axes.0140

And that is one way to remember what a root is.0150

How do we find the roots of a polynomial? Well, at first we might try a naive approach and attempt to solve the way we are used to.0153

Naive is just what you have done before--what seems to make sense--without ever really having had a whole lot of experience about it.0160

So, the naive attempt would be probably to just isolate the variable on one side.0166

That is what we did with a bunch of other equations before, so let's do it again.0170

Now, in some cases, this will actually work, and we will find all of the real solutions.0173

For example, if we have f(x) = x - 3, then we set it equal to 0, because we are looking for the roots; we are looking for when 0 = x - 3.0176

We move that over, and we get x = 3; great.0184

Or if we had y = x3 + 1, we set that to 0, and we have 0 = x3 + 1,0187

because we are looking for what x's cause us to have 0 = x3 + 1.0192

So, we have x3 = -1; we take the cube root of both sides, and we get x = 3√-1, which is also just -1.0197

So, in both of these cases, the naive method of isolating for variables worked just fine.0204

But that is definitely not going to be the case for all situations.0208

The naive method of isolation will fail us quite quickly, even when used on simple quadratic polynomials.0212

Consider f(x) = x2 - x - 2--that is not a very difficult one.0217

But this method of trying to isolate will just fail us utterly if we use it here.0223

So, 0 = x2 - x - 2...we might say, "Well, let's get the numbers off on one side."0227

We have 2 = x2 - x; but then, we don't have just x; so let's pull out an x; we get 2 = x(x - 1).0232

And well, we are not really sure what to do now; so let's try another way.0240

0 = x2 - x - 2; let's move x over, because we are used to trying to get just x alone.0243

So, we have x here; but then we also have x2 here; so let's divide by x.0248

We get 1 = (x - 2)/x; once again, we are not really sure what to do.0252

Let's try again: 0 = x2 - x - 2, so let's move over everything but the x2.0256

Maybe x2 is the problem; so we will get x + 2 = x2.0262

We take the square root of both sides; we remember to put in our ± signs: ±√(x + 2) = x.0265

I don't really know how to figure out what x's go in there to make that true.0271

So, in all three of these cases, it is really hard to figure out what is going on next.0274

If we are going to try to isolate, we are going to get these really weird things.0280

This method of isolation that we are used to isn't going to work here, because we can't get x alone; we can't get the variable alone on one side.0283

It is not going to let us find the roots of polynomials if we try to isolate.0290

But at least we could trust it in those previous examples; we saw that we can trust it when it does work.0293

So, we might as well try it first--no, it is even worse: the naive method of isolation0298

can make us miss answers entirely, even though we think we will know them all.0303

So, we will think we have found the answers; but in reality, we will only have found part of the answers.0307

Consider these two ones: if we have 0 = x2 - 1, we move the 1 over; and then we take the square root of both sides.0311

The square root of 1 is 1; the square root of x2 is x; great.0317

For this one over here, we have 0 = p2 + 3p, so we realize that we can divide both sides by p.0320

And since 0/p is just 0, we have 0 = p2/p (becomes p); 3p/p becomes 3; so we have 0 = p + 3.0327

We move the 3 over; we get p = -3; great--we found the answers.0336

Not quite: those above things are solutions, but in each case, we have missed something.0340

We have been tricked into missing answers by trying to follow this naive method.0345

The other solutions for this would be x = -1 and p = 0, respectively.0349

The mistakes that we forgot were a ± symbol over on this one; we forgot to put a ± symbol when we took the square root;0353

and then, the other one was dividing by 0, because when we divided, we inherently forgot0361

about the possibility that, if we were actually dividing by 0, we couldn't divide by 0.0366

So, those are the two mistakes; but even if you personally wouldn't have made those same mistakes,0371

this example shows how it is easy to forget those things in the heat of actually trying to do the math.0375

You might forget about that; you might accidentally make one of these mistakes; so it is risky to try this method of isolation.0381

We need something that works better.0387

We find the roots of the polynomial by factoring; we break it into its multiplicative factors.0389

Let's look at how this works on the example that we couldn't solve with naive isolation.0395

We have f(x) = x2 - x - 2; we have 0 = x2 - x - 2, because we are looking for when f(x) is equal to 0.0399

And then, we say, "Let's factor it; let's break it into two things."0407

So, we have (x - 2) and (x + 1); and if we check, that does become it: x times x becomes x2; x times 1 becomes + x;0409

-2 times x becomes - 2x; -2 times 1 becomes -2; so yes, that checks out to be the same thing as x2 - x - 2.0417

0 = x - 2 and 0 = x + 1 is how we now set these two things equal to 0; we have 0 = x - 2 and 0 = x + 1.0425

And we get x = 2 and x = -1, and we have found all of the solutions for this polynomial.0435

Why does this work, though? We haven't really thought about why it works.0441

And we don't want to just take things down and automatically say, "Well, my teacher told me that, so that must be the right thing."0444

You want to understand why it is the right thing.0449

Teachers can be wrong sometimes; so you want to be able to verify this stuff and say, "Yes, that makes sense,"0451

or at least have them explaining and saying, "Well, we don't understand quite enough yet;0456

but later on you will be able to see the proof for this"; you really want to be able to believe these things,0460

beyond just having someone tell you by word of mouth.0463

So, to figure out why this has to be the case, we will consider 0 = ab.0466

The equation is only true if a or b, or both of them, is equal to 0; if neither a nor b is equal to 0, then the equation cannot be true.0473

If a = 2 and b = 5, then we get 10, which is not equal to 0.0482

As long as a or b is 0, it will be true, because it will cancel out the other one.0489

But if both of them are not 0, then it fails, and it is not going to be 0.0494

It is the exact same thing happening with x2 - x - 2; we have 0 = x2 - x - 2,0499

which we then factor into (x - 2) and (x + 1); so let's use two different colors, so we can see where this matches up.0505

We are pairing this to the same idea of the a times b equals 0; it is (x - 2) (x + 1) = 0.0513

The only way that this equation can be true is if x - 2 = 0 or x + 1 = 0.0520

Just as we showed up here, it has to be the case that a or b equals 0 for that to be true.0528

So, it must be the case that either (x - 2) or (x + 1) equals 0, if this is going to be true.0534

So, our solutions are when either of the two possibilities is true--if the possibility is true, if one of them is true, then the whole thing comes out.0539

So, either case being true makes it acceptable; that gets us 2 (keeping with our color coding) and -1 as the two possibilities.0546

So, by breaking x2 - x - 2 into its factors, we can find its roots.0557

So, this is how we find polynomial roots, in general: the first thing we do is set the whole thing as 0 = polynomial.0562

We have to have some polynomial, and then it is 0 equals that polynomial.0569

The next thing we do is factor it into the smallest possible factors; we break it down into multiplicative factors.0573

And then finally, we set each factor equal to 0, and we solve for each of them.0580

So, in step number 2, we are going to get things like 0 = (x + a)(x + b)(x + c)(x +d) and so on, and so on, and so on.0585

And then, in step 3, we set each factor to 0; so we get things like x + a = 0, at which point we can solve and say, "Oh, x = -a."0595

That is one of our possible solutions; and from there, you can work out all of the roots of the polynomial.0604

Caution--this is very important: notice that it is extremely, extremely important to begin by setting the equation as 0 = polynomial.0610

I have seen lots of mistakes where people forgot to set it as 0 = polynomial.0618

If it isn't, if it was something like 5 = (x - 2)(x + 1), we can't solve for the solutions from those factors.0621

Those factors are now meaningless; they aren't going to help us.0628

We need the special property that 0 turns everything it multiplies into 0.0631

Without that special property, this method just won't work.0635

Consider if we had something like 5 = ab; there is no way that we could just figure out what the answers are here.0638

It is not just simply that a has to be 0 or b has to be 0, because a could be 5 and b could be 1.0644

Or b could be 5 and a could be 1, or a could be 25 and b could be 1/5; or a could be 100 and b could be 1/20.0649

We have lots of different possibilities--a whole spectrum of things; there are way too many possible solutions.0658

We need that special property of 0 = ab to be able to really say, "That thing is 0, or that thing is 0"; that is what we know for sure.0663

That is how we get useful information out of it.0671

That is why it has to be 0 = polynomial; if you don't set it up as that before you try to factor it,0674

before you try to do the other steps, you are just not going to be able to get the answer,0679

because we need that special property that 0 has when it multiplies other things.0682

0 multiplying something automatically turns it to 0; if we don't have that special property, things just won't work.0686

Factoring is not necessarily easy: say we have something like0693

x5 + 6.5x4 - 17x3 - 41x2 + 24x, and we want to know what the zeroes are.0697

Well, if we knew its factors, we would be able to break it into (x + 8)(x + 2) times x times (x - 0.5) times (x - 3).0703

And it would be really easy to figure out what the polynomial's roots are.0710

At this point, we say, "Well, great; x + 8 becomes x = -8; x + 2 becomes x = -2; x + 0 becomes x = 0;0712

x - 0.5 becomes x = positive .5; and x - 3 becomes x = positive 3--great; I found it; it is really easy to find its roots."0722

But how do you factor a monster like that?0732

Ah, there is the problem: factoring can be quite difficult.0734

Luckily, by this point, you have been certainly practicing how to factor for years in your algebra classes.0738

By now, you have done lots of factoring; you are used to this; you have played with polynomials a bunch in previous math classes.0743

And all that work has a use, and it is here, finding roots; we can break things down into their factors and find roots.0748

But there is no simple procedure for factoring polynomials.0754

Once again, remember: if you were confronted by something like this, you would probably have a really difficult time0757

figuring out what its factors were--figuring out how you can break that down into factors.0762

There is no simple procedure for factoring polynomials.0768

High-degree polynomials can just be very difficult to factor; happily, we are not going to really see such polynomials.0770

Most courses on this sort of thing don't end up giving you very difficult things to factor at this stage.0777

So, we won't really have to worry about factoring really difficult polynomials.0782

We will be able to stick to the smaller things.0787

So, let's have a quick review of how you factor small things like quadratics.0790

We are going to see a bunch of quadratics; they are very important--they pop up all the time in science.0795

So, let's look at a brief review of how to factor a quadratic polynomial.0799

Remember, quadratic is a degree 2, and a trinomial just means 3 terms.0801

So, if we have a quadratic trinomial in its normal form, then we have ax2 + bx + c.0806

If we want to factor that, we will turn it into a pair of linear binomials: degree 1 and 2 terms (linear and binomial).0811

We would want to break this into (_x + _) (_x + _).0819

Great; to be equivalent to the above, the coefficients of x must give a product of a.0824

So, a has to come out with the red dot here, times the red dot here.0830

And the constants have to give a product of c; so the blue dot here times the blue dot here has to come out to be c.0835

Also, b has to add up from the products of the outer blanks and the inner blanks.0843

So, it has to be that the red dot here times the blue dot here, plus the blue dot here times the red dot here, comes up to be this b here.0848

So, it is mixed out of the two of them.0863

Don't worry if that is a little bit confusing right now; you will be able to see it as we work through examples.0865

The b has to add up to the products of the outer blanks and the inner blanks.0868

The a has to come from the first blanks, and the c has to come from the last blanks.0872

Don't worry about memorizing this, though--it is just a sense of what is going on in practice.0876

Let's look at an example: we want to factor 2x2 - 5x - 12.0880

So, we know, right away, that we want to break it into the form (_x + _)(_x + _).0884

The first thing we notice is that we have this 2 at the front; and 2 only factors into 2 times 1.0890

We can't break it up into anything else really easily; so let's put 2 times 1 down as 2x times x.0895

We have to put the 2 somewhere; so it is either going to be (2x + _)(1x + _), or it is going to be the 2 over here and the 1 here.0901

It doesn't really matter what order we put it in; so we will put the 2 at the front.0910

We have (2x + _)(x + _); now, what is going to go into those other blanks?0913

Now, we need to factor the -12; so let's factor 12 first: we notice that 12 can break into 1 times 12, 2 times 6, or 3 times 4.0917

And one of the factors has to be a negative, because we have a negative in front of the 12.0926

So, they have to be able to multiply to make a -12; so there is going to have to be a negative on either the 1 or the 12.0931

One of those two will have to have a negative on it; we don't know which one, but it is going to be one of them.0938

Or it is going to be -2, or -6; and then finally, for the last pair, it would be -3 or -4.0943

I am not going to use all of these at once; we will have to figure out which one is right.0948

But one of them will have to be negative, because of this negative sign up here.0950

We start working through this; and we know that there is this 2x here at the front.0955

We have this 2x at the front; so it is going to multiply this one, and it is going to effectively double whatever we put here.0960

So, the difference between one of the numbers doubled and its sibling (the one here times this one in front of it) must be -5, because we get -5x.0967

We notice that 3 - 2(4)...2 times 4 is 8; 3 is 3; so the difference between those is 5.0979

So, we can set it up as 3 - 2(4) = -5; and we get (2x + 3)(x - 4); and we have factored this out; we have been able to work it out.0986

Now, there are various ways, various tricks that have been taught to you in previous classes.0998

But the important thing is just to set up and have an expectation of what form you are trying to get.1005

And then, plug things in and say, "Yes, that would work; that would get me what I am looking for" or1009

"No, if I plug that in, that won't work; that won't get me what I am looking for."1013

As long as you work through that sort of thing, you will be able to find the answer eventually.1016

It is always a good idea, though, to check your work; you will find the answer, but it is really easy to make mistakes.1020

So, even on the easiest of problems (like the one we were just working on), there are lots of chances to make mistakes; trust me.1026

I make mistakes; everybody makes mistakes; the important thing is to catch your mistakes before they end up causing problems.1032

So, it means that you should always try expanding the polynomial after you have factored it, to make sure you factored correctly.1038

And it is OK to do this in your head; once you get comfortable with doing this sort of thing1044

(and by now, honestly, you probably have had enough experience with this that you can just do this in your head reasonably quickly),1048

it is OK to do it in your head; the important part is that you want to have some step where you are checking back on what you are doing.1053

So, either do it on the paper (if it is a long one) or do it in your head (if it is something that is short enough, easy enough, for you to check).1058

But you want to make sure that you are checking your work.1063

For example, if we have 2x2 - 5x - 12, we figured out that that breaks into (2x + 3) and (x - 4).1065

But we want to check and make sure that it is right.1071

So, we check and make sure: 2x + 3 times x - 4...we get 2x2, and then 2x(-4), 3x...we get -8x + 3x...1073

3 times -4...we get -12; we combine our like terms of -8x + 3x, and we get 2x2 - 5x - 12.1083

Sure enough, it checks out; we have what we started with, so we know that our factoring was correct; we did a good job.1092

Factoring higher-degree polynomials: in general, factoring polynomials of any degree is going to be similar to what we just did on these previous few slides.1100

The only difference is that it will become more complex as they become longer, as we get to higher and higher degrees.1107

So, for example, if we had something like a cubic--if we had ax3 + bx2 + cx + d--1112

we would probably want to set it up as (_x + _)(_x2 + _x + _).1118

If we are going to be able to break it up and factor it, it is going to have to factor into these two things, (_x + _)(_x2 + _x + _).1123

Notice also that this is a degree 1, and this is a degree 2; and when you multiply these two together, you will get back to a degree 3 over here.1131

Adding up the degrees on the right side has to be what we had on the left side.1141

We could also work on quartic, a degree 4; and we might try one of these two templates.1145

If we have ax4 + bx3 + cx2 + dx + e, we might break it1149

into (_x + _)(_x3 + _x2 + _x + _); or we might break it into (_x2 + _x + _)(_x2 + _x + _).1153

And so on and so forth...clearly, this is going to become more difficult as we get to higher and higher degrees.1163

The higher the degree of a polynomial, the more complicated our template is going to have to be for where we are going to fit things in.1168

The more choices we are going to have, the more difficult it is going to be to do this.1173

Luckily, we are only occasionally going to need to factor cubics, things like this where it is degree 3.1177

And we are very, very seldom going to see anything of higher degree.1182

So, don't worry too much about having really difficult ones.1186

But just be aware that factoring really large, high-degree polynomials can actually be pretty difficult to do.1188

Roots imply factors: we have this useful trick if we want to break out these higher polynomials.1196

If we know one of a polynomial's roots, we automatically know one of its factors.1201

Remember: one use of finding the factors of a polynomial is to find its roots.1206

If you find a factor of the form (x - a), then you set that equal to 0, and you know that that is going to be x = a.1209

It turns out that the exact opposite is true; if we know a polynomial has a root at x = a, then it also means we have a factor of x - a.1216

So, if we have a root x = a, then that turns into a factor, x - a, just because of this equation here,1225

where we are setting that factor equal to 0, which gives us the root.1232

We won't prove this; it requires a little bit of difficult mathematics, and some things that we actually haven't covered in this course yet.1236

But we can see it as a theorem: Let p(x) be a polynomial of degree n; then if there is some number a,1242

such that p(a) = 0 (that is to say that a is a root of our polynomial, p--if we plug a in, we get 0),1250

then there is some way to break it up so that it is p(x) = (x - a), our factor (x - a) that we know1257

from our root at x = a, times q(x), where q is some other polynomial of degree n - 1,1264

because this here is degree 1; so when we multiply it by a degree n - 1, we will be back up to our degree n polynomial that we originally had.1272

If we manage to find one or more roots, sometimes the problem will give them to use; other times, we will get lucky, and we might just guess one.1285

This theory means we automatically know that many factors of the polynomial.1291

For example, say we know that p(x) = x3 - 2x2 - 13x - 10.1295

And we are told that p(5) = 0; we know that 5 is a root.1300

Then, we automatically know that at x = 5 we have a zero; so (x - 5) is our root.1305

We plug that in; we know it is going to be (x - 5)(_x2 + _x + _).1312

Now, we don't know what is in these blanks yet; we still don't know what is going to be in there.1317

But we are one step closer to figuring out what those factors are, for being able to figure out what has to go in those blanks.1325

Later on, in another lesson, we will use this fact to great advantage--1334

this fact that knowing the root automatically means you know a factor--1336

When we learn about the intermediate value theorem to help us find roots,1340

and the polynomial division using those roots to break down large polynomials into smaller, more manageable factors.1344

Not all polynomials can be factored, though; even with all of this talk of factoring polynomials, there are some that cannot be factors,1352

not because it is difficult or really hard to do, but because it is just simply impossible.1358

Consider this polynomial: f(x) = x2 + 1; if we try to find its roots, then we have 0 = x2 + 1.1363

So, we have x2 = -1; but there is no number that exists that can be squared to become a negative number.1370

No number can be squared to become a negative number; why?1377

Consider: if we have (-2)2, that becomes positive 4; if we square any negative number, it becomes a positive number.1380

If we square any positive number, it stays positive; if we square 0, it stays 0.1386

So, there is no number that we have, that we can square, and get a negative number out of it.1391

Thus, the polynomial has no roots; and since it has no roots from that theorem we just saw, it can't have any factors.1395

So, it has no roots; therefore, it cannot be reduced into smaller factors.1401

And something that cannot be reduced, we call irreducible; it is not reducible.1407

Now, I will be honest: what I just told you isn't really the whole story.1413

More accurately, we can't factor all polynomials yet.1418

The previous slide that we just saw is perfectly true, but only if you are working with just the real numbers (which have the symbol ℝ like that).1422

Now, that is what we are normally working with; so it is kind of reasonable to say this.1432

But it turns out that there is a hidden type of number that we haven't previously explored.1435

You might have seen this in previous math classes, even.1439

We will learn about the complex numbers later on; complex numbers can give us a way to factor these supposedly irreducible polynomials.1441

So, they are irreducible for real numbers; but they are not irreducible for complex numbers.1449

Now, we will learn about them in the lesson that is named after these numbers--our lesson on complex numbers.1454

But for now, we are just working with real numbers; and in general, we will just be working with real numbers in this course.1460

Real numbers are really useful; you can do a lot of stuff with them.1464

So, it is enough for us to be working with real numbers, generally.1467

That means, for us right now, at least some polynomials are simply irreducible.1470

And we can't always find roots for everything in a polynomial, because we can't break it down,1474

because there are things that just don't have roots, based on how real numbers work.1480

Now, we will talk about complex numbers later on; but I just wanted to point this out--1484

that I am not telling you the whole story right now, because we don't want to get confused with complex numbers.1488

But for our purposes right now, with real numbers, there are some things that are simply irreducible.1492

There is a limit to how many roots or factors a polynomial can have.1498

Now, roots and factors are basically two sides of the same thing.1501

Since x = a as a root is the same thing as knowing (x - a) as a factor, they are just two sides of the same thing.1505

So, we will consider them as roots/factors of polynomials.1516

A polynomial of degree n can have, at most, n roots/factors; so we can have a maximum of n of these.1518

Why is this the case? Well, consider this: every factor comes in the form (x + _), or even larger if the factor is irreducible.1527

It might be (_x2 + _x + _); but the very smallest has to be (x + _).1535

If we break a polynomial into its factors, we are going to get (x + _) (x + _)...so on and so on, up until (x + _).1540

Now, if we had more than n factors, then that would mean that we have (x + _) multiplying by itself more than n times.1547

So, if we have x multiplying more than n times, then it has to have a degree larger than n; its exponent is going to have to be greater than that.1555

If we wanted to max out at x2, but we had (x + _)(x + _)(x + _)...well, that is going to become x31564

plus stuff after it, which is not going to be x2...not going to be degree 2.1576

So, if we have a degree n polynomial, the most factors we can possibly have are n factors, n roots,1581

because otherwise, we would have too many factors, and we would blow out the degree of the polynomial.1589

Thus, the most roots/factors a polynomial can have is equal to its degree.1595

We also can get information about the possible shape of a polynomial's graph from its degree.1601

A polynomial of degree n can have, at most, n - 1 peaks and valleys.1605

Formally speaking, that is relative maximums and minimums.1610

For example, if we have x4, then that means we have n = 4 (our degree).1613

So, n - 1 = 3; so we look over here, and we have one valley here, one peak here, one valley here.1618

This is also a relative minimum, a relative minimum (that is what we mean by "valley), and a relative maximum (that is what we mean by "peak").1630

So, n - 1 tells us the most bottoms and tops we can have, before they either go off to positive infinity or go off to negative infinity.1642

Now, we can't prove this here, because it requires calculus; but it is connected with the maximum number of roots in a polynomial.1650

And if you go on to take calculus (which I heartily recommend), you will very clearly, very quickly see it.1656

It becomes very clear in calculus; it is one of the important points of what you do in calculus.1661

So, you will think, "Oh, that makes a lot of sense," because the possible peaks and valleys1665

are connected to a polynomial that has a degree that is 1 less; and that is why it is connected.1670

Don't worry about it too much right now; but it is very interesting, and very obvious, if you go on to take calculus.1675

Notice that, in both of the previous properties, it was described as "at most."1681

Just because a polynomial has degree n does not mean it will have n distinct roots or n - 1 peaks and valleys.1687

We aren't necessarily going to have to have that many; it is just that we can have up to that many.1693

Consider f(x) = x5 + 1; this graphs like this, but from this graph, we can see clearly:1697

we only have one root, and we have no peaks or valleys.1704

The degree gives an upper limit on how many there can be, but it doesn't tell us how many there will be.1708

It just that the maximum is this; but you could definitely have fewer.1713

All right, we are ready for some examples.1719

The first one: we want to find the zeroes of f(x) = 3x2 - 23x + 14.1721

So, this is a textbook example--literally a textbook example, since this is effectively a textbook.1726

So, you plug in 0, because we are looking for when f(x) is equal to 0.1731

0 = 3x2 - 23x + 14; we know we are going to be looking for 0 =...something where it is going to be (_x + _)(_x + _).1735

What are we going to slot in there? Well, we notice that here is 3; the only way we can break up 3 is 3 times 1.1751

There are no other choices; so we either have to have 3 go for the first x or 3 go to the second x.1758

So, let's set it as 0 = (3x + _); and we will have 1x, so just x, plus _.1762

Great; now, at this point, we also say, "We have 14 over here; how can 14 break up?"1772

Well, we can have 1 times 14, or we can have 2 times 7; those are the only choices.1777

So, we are going to have to plug in either 1 times 14 or 2 times 7.1783

But now, we also have to take this -23 into consideration.1786

If we have -23, then we are going to have at least one negative over here.1789

And since it comes up as a positive, it is going to have to be that they are both negatives.1793

So, one of them is negative, so they are going to both be negatives; so it will be -1(-14) or -2(-7).1796

So, -1 times -14...we will notice that, either way we put that in, that won't work out.1802

But we can plug in -2 times -7, and we can amp up this -7; so + -7 here...put in the -2 here...and we get 0 = (3x - 2)(x -7).1807

We have managed to factor it; let's really quickly check what we have here--does this work out?1823

Check (3x - 2)(x - 7); we would get 3x2 - 21x - 2x + 14, so 3x2 - 23x + 14; it checks out; sure enough, it is good.1828

So, at this point, we break this down into two different possibilities: either 3x - 2 = 0, or x - 7 = 0.1847

So, 3x - 2 = 0 or x - 7 = 0 are the two different worlds where this will be true,1855

where we will have found a root where the whole expression will be equal to 0.1861

So, 3x - 2 = 0: we get 3x = 2; x = 2/3; over here, x - 7 = 0; we have x = 7; so our answers are x = 2/3 and x = 7; those are the roots for this polynomial.1864

Great; if f(2) = 0, factor f(x) = x3 - 7x + 6.1881

Remember: if we know that at x = 2 we have a zero (at x = 2 there is a root), then that means there is a factor in that polynomial of (x - 2).1888

How do we figure that out? Well, we notice that x = 2; then it is x - 2 = 0, so that implies that it has to be a factor of (x - 2) in there.1904

We can use that piece of information; we know that f(x) is going to have to break down with an (x - 2) in there.1912

So, let's set it up like normal: 0 = x3 - 7x + 6; but what we just figured out here...we know that there is a factor of (x - 2).1917

So, we can also write this as 0 = (x - 2)(_x2 + _x + _); what are going to go into those blanks?1924

Well, at this point, we just use a little bit of logic and ingenuity, and we can figure this out.1935

Well, we know that...what is in front of this x3? It is effectively a 1.1941

So, if there is a 1 in front, we have x times x2; whatever goes into this blank is going to determine what coefficient is in front of it.1944

So, since we want a 1, it has to be that there is a 1 here, as well.1952

What about the very end? Well, the only thing that is going to create the ending constant is going to be the other constants.1957

So, the constants that we have here are -2 and whatever goes into that blank; so it must be that -2 times _ here becomes 6.1968

-2 times -3 becomes 6; so we have a -3 here.1975

And finally, what is going to go into this blank here?1980

We think about this one, and we know that we want to have 0x2 come out of this; there are no x2's up here.1984

So, we have + 0x2; so whatever we put into this blank must somehow get us a 0x2 to show up.1992

So, x times x2 is x3; so we are not going to worry about that.2001

But x times _x is going to be x2; let's do a little sidebar for this.2005

x times _x will become _x2; and -2 times...we already filled in that blank...1x2 is going to be -2x2.2010

Now, we want the 0x2 out of it; so it must be that, when we add these two things together, it comes out to be 0x2.2023

What does this have to be? It has to be positive 2x2.2031

We know that positive 2x2, minus 2x2, comes out to be 0x2; so it must be that this is a positive 2x.2035

So, we write this whole thing out: 0 = (x - 2)(x2 + 2x - 3); and we have been able to figure out that that works.2042

We check this out and do a really quick check; so x3 + 2x2 - 3x - 2x2 - 4x - 6...2053

x3 checks out; 2x2 - 2x2 cancel each other; that checks out.2067

-3x - 4x; that becomes -7x, so that checks out; -6...that checks out as well.2072

Great; we have a correct thing, so 0 = (x - 2)(x2 + 2x - 3) is correct.2078

We factored it properly; so at this point, the only thing that we have left to factor is the x2 + 2x -3.2087

0 = (x - 2)(_x + _)(_x + _); what goes in those first blanks?2095

Well, we just have a 1 in front of that; so it is 1 and 1...we don't have to worry about it that much.2103

What else is going to go in there? Well, -3 is at the very end; we have +2x, so it must be that the negative amount is smaller than the positive amount.2108

So, it is going to be + -1 and + 3; -1 times 3 gets us -3, and everything else checks out.2117

x times x is x2; plus 3x minus x...that gets + 2x; and minus 1 times 3 gets us -3; so that checks out.2128

We did another check in our heads really quickly.2136

So finally, we have 0 = (x - 2)(x - 1)(x + 3); we break this up into three different worlds;2138

set each world equal to 0: x - 2 = 0; x - 1 = 0; x + 3 = 0; so we have x = 2; x = 1; x = -3.2152

Those are all of the roots for this polynomial.2167

All right, the next example: give a polynomial with roots at the indicated locations and the given degree.2172

Now remember, a root can become a factor; so if we know that we have a root at -3, then that becomes a factor of...2177

if it was x = -3, then it becomes (x + 3); if we had x = 8, then that would become (x - 47)...2186

I'm sorry; I accidentally read the wrong one--read forward one; that is (x - 8).2196

And then finally, if we had x = 47, we would have (x - 47) as our factor.2201

So, those three things together...we have (x + 3)(x - 8)(x - 47), and that right there is a polynomial.2207

We know it has degree 3, because we have x times x times x; that is going to be the largest possible exponent we can get on our variable.2217

That will come out to be x3, so we have a degree 3; and we know it has roots in all of the appropriate places.2226

And we are done--that is it; we could expand this, and we could simplify, if we wanted.2230

We weren't absolutely required to by the problem, and this is a correct answer.2234

It is a polynomial; it is not in that general, standard form that we are used of _x to the exponent, _x to the exponent, _x to the exponent.2239

But it is still a polynomial, so it is a pretty good answer; we will leave it like that.2248

The next one: we have -2 and positive 2, so we have (x + 2) for the -2 and (x - 2) for that.2252

Why? x = -2; we move that over; we get x + 2 = 0; x = +2...we move that over and we get x - 2 = 0.2261

So, we get (x + 2)(x - 2); but if we multiply those two together, we just have a degree of 2, and we want a degree of 4.2268

So, we need to somehow get the degree up on this thing, but have the same roots--not have to accidentally introduce any more roots.2275

So, if we introduced multiplying just by x twice, we would have introduced a root at x = 0.2284

So, we can't just do that; but we do realize that if we just increase this to square it on both of them, they will still have the same roots.2289

It is just duplicate roots showing up; so (x + 2)2 + (x - 2)2...we have hit that degree of 4,2296

because each one of these will now have a degree of 2.2302

Alternately, we could have done this as (x + 2) to the 1, (x - 2) cubed, or (x + 2)3(x - 2)1.2305

Any one of these would have degree 4, and have our roots at the appropriate place.2315

Great; the final example: What is the maximum possible number of roots and peaks/valleys for each of the following polynomials?2320

So, for our first one, f(x), we notice that this has a degree of 3; so n = 3 means the maximum number of roots it can have is 3,2327

and the maximum peaks/valleys is one less; n - 1 is 3 - 1, is 2.2339

So, the maximum number of roots is 3; the maximum peaks/valleys is 2.2350

We don't necessarily know it will have that many; all we know is that that is the maximum it could possibly have.2354

The next one: we notice that the degree for this one is 47; so if n = 47, then the maximum roots are going to be equal to that degree.2359

The maximum peaks/valleys are going to be one less than that degree, so we will get 46, one less than that.2369

The final one: for this one, we think, "Oh, 103, so it is 3!"--no, we have to remember that this is not a variable.2379

This here is a variable; so it is x1, so its degree is just 1.2388

For that one, degree 1...we will change over to the color green...n = 1 means the maximum roots are just 1;2394

and the maximum peaks/valleys are going to be one less than 1, so 1 - 1 = 0.2405

Now, why is that the case? Well, think about it: 103 - 5...10 cubed is just some constant;2415

it happens to be 1,000, but that is not really the point; so 1000x - 5 is just going to be a very steep line.2420

x - 5...it will intersect here; but does it ever go up and down--does it ever undulate in weird ways?2429

No, it never does anything; we just have a nice, straight line, since it is a linear thing (linear like a line).2435

So, since it is a linear expression, it never undulates--never has any peaks or valleys.2441

So, it never has any relative maximums, and no relative minimums; and that is why we have 0 there--it makes sense.2445

All right, I hope everything there made sense; I hope you got a really good understanding of roots,2451

because roots will come up in all sorts of places; they are really important to understand.2454

It is really important to understand this general idea, because you will see it in other things, being changed around.2457

But if you understand this general idea, you will be able to understand what is going on in later things and different courses.2461

All right, see you at Educator.com later--goodbye!2466