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

Partial Fractions

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

 2 answersLast reply by: Peter KeTue Nov 10, 2015 8:11 PMPost by Peter Ke on November 8, 2015At 11:40 "Quick Example", I understand at the left side of the equation, how you got rid of the denominator by multiplying but then how you got the right side though?The right side was A(x^2 - x + 2) + (Bx+C)(x+3). How did you get this!? I really don't get it... 1 answerLast reply by: Professor Selhorst-JonesTue Sep 9, 2014 9:52 AMPost by David Llewellyn on September 8, 2014Your explanation that an expression with all exponents of x <=0 does not qualify as a polynomial (see my question in Polynomials lecture) explains why you stop the polynomial division with a remainder once you hit a constant in this lecture. BTW, presumably, you can also have irreducible factors of any even power (x^2n) n>= 1  with 2n terms in the denominator. I can see that all polynomials will have, at least, one real factor as this polynomial will generate an odd function.

### Partial Fractions

Note:Before starting, it should be noted that this is a rather difficult concept to explain just with writing. It helps a lot to see it in action, so it's strongly recommended that you watch the video if you find any of this confusing.
• Long ago, you learned how to combine fractions: you put them over a common denominator, then combine them. But what if we wanted to do the reverse? What if we had a fraction that we wanted to break up into multiple, smaller fractions? We call these smaller fractions partial fractions and the process partial fraction decomposition.
• To understand this lesson, you'll need some familiarity with solving systems of linear equations. Previous experience from past Algebra classes will probably be enough, but if you want a refresher, check out the lesson Systems of Linear Equations. Knowing how to factor polynomials and understanding them in general will also be necessary, along with the ability to do polynomial division.
• If we have a polynomial fraction of the from [N(x)/D(x)], there are two possibilities in regards to the degrees of the top and bottom polynomials:
• Proper: degree of N(x) < degree of D(x);
• Improper: degree of N(x) ≥ degree of D(x).
To decompose a polynomial fraction, it must first be proper. If the fraction is improper and we want to decompose it, it must first be made proper through polynomial division. [Remember that the remainder from the division goes back on the original denominator, which we will then decompose.]
• Once the polynomial fraction is proper, the next step is to factor the denominator. After it's broken down into its smallest factors, we're ready to decompose. Notice that there are two types that these smallest possible factors can come in:
• Linear factors raised to a power: (ax+b)m;
• Irreducible quadratic factors raised to a power: (ax2+bx+c)m. [Remember, `irreducible' means it can't be broken up further. That means quadratics like x2+1 or 5x2−3x+20.]
• The partial fraction decomposition must include the following for each linear factor (ax+b)m:
 A1ax+b + A2(ax+b)2 + …+ Am(ax+b)m ,
where A1, …, Am are all constant real numbers.
• The partial fraction decomposition must include the following for each irreducible quadratic (ax2+bx+c)m:
 A1x+B1ax2+bx+c + A2x+B2(ax2+bx+c)2 + …+ Amx+Bm(ax2+bx+c)m ,
where A1, …, Am and B1, …, Bm are all constants.
• If we have multiple of a given type or mixed types, we just decompose each of them based on their rules and put them all together.
• Finally, we need to figure out what goes on the numerators. To do this, have the original fraction on one side of an equation and the partial fraction decomposition on the other. Then multiply both sides of the equation by the denominator of the original fraction. This gives an equation that we can transform into a system of linear equations and solve to find the values for each of the constants.

### Partial Fractions

In the context of rational functions, what does it mean for a fraction to be proper? To be improper? How does this connect to partial fraction decomposition?
• A rational function is a fraction made out of polynomials: the numerator and the denominator are both polynomials. This means we can talk about the degree (largest exponent on a variable) of each of the two polynomials.
• A fraction is proper when the degree of the numerator is less than the degree of the denominator.
• A fraction is improper when the degree of the numerator is equal to or greater than the degree of the denominator.
• These two definitions matter for partial fraction decomposition because only proper fractions can be decomposed. If a fraction is in improper form, it must first me put into proper form through polynomial division. Only then can it be decomposed.
Proper: the numerator's degree is less than the denominator's. Improper: the numerator's degree is equal to or greater than the denominator's. This connects to partial fraction decomposition because only proper fractions can be decomposed.
Write the form of the partial fraction decomposition, but do not solve for the constants.
 3x−2 (x+1)(x−3)
• To decompose a fraction, the fraction must be proper. For this problem, we see that the numerator has a smaller degree than the denominator, so we're fine to continue. Next, we need to factor the denominator completely. This has already been done for us.
• Once the denominator is factored, we can create an equation where the left-hand side is the original fraction and the right-hand side uses the denominator's factors and currently unknown constants. [See the lesson if you're not sure how to do this; it is fully explained in the lesson and much easier to understand visually.]
 3x−2 (x+1)(x−3) = A x+1 + B x−3
[(3x−2)/((x+1)(x−3))] = [A/(x+1)] + [B/(x−3)]
Find the partial fraction decomposition.
 2x+6 x(x+2)
• To decompose a fraction, the fraction must be proper. For this problem, we see that the numerator has a smaller degree than the denominator, so we're fine to continue. Next, we need to factor the denominator completely. This has already been done for us.
• Once the denominator is factored, we can create an equation where the left-hand side is the original fraction and the right-hand side uses the denominator's factors and currently unknown constants. [See the lesson if you're not sure how to do this; it is fully explained in the lesson and much easier to understand visually.]
 2x+6 x(x+2) = A x + B x+2
• To solve for the constants on top of the fractions, multiply both sides by the factors in our starting fraction's denominator:
 x(x+2) · ⎡⎣ 2x+6 x(x+2) = A x + B x+2 ⎤⎦

 2x+6 = A(x+2) + B(x)

 2x+6 = Ax + Bx + 2A
• From here, we can solve based on each type. The only things that can contribute to the number of x's are terms with an x on them, and the only thing that can contribute to the constant number are terms with no variables on them. Thus, we can split the above into two equations:
 2x = Ax + Bx               6 = 2A
• We can easily solve the right-side equation and get A = 3. Once we know that, we can plug it into the other equation to find B:
 2x = 3x + Bx,
so B=−1.
• Now plug our values of A=3 and B=−1 into the decomposition we set up near the beginning to find how the fraction breaks down:
 2x+6 x(x+2) = A x + B x+2 = 3 x + −1 x+2
[3/x] + [(−1)/(x+2)]
Find the partial fraction decomposition.
 4x−3 (x−6)(x+1)
• To decompose a fraction, the fraction must be proper. For this problem, we see that the numerator has a smaller degree than the denominator, so we're fine to continue. Next, we need to factor the denominator completely. This has already been done for us.
• Once the denominator is factored, we can create an equation where the left-hand side is the original fraction and the right-hand side uses the denominator's factors and currently unknown constants. [See the lesson if you're not sure how to do this; it is fully explained in the lesson and much easier to understand visually.]
 4x−3 (x−6)(x+1) = A x−6 + B x+1
• To solve for the constants on top of the fractions, multiply both sides by the factors in our starting fraction's denominator:
 (x−6)(x+1) · ⎡⎣ 4x−3 (x−6)(x+1) = A x−6 + B x+1 ⎤⎦

 4x−3 = A(x+1) + B(x−6)

 4x−3 = Ax + Bx + A−6B
• From here, we can solve based on each type. The only things that can contribute to the number of x's are terms with an x on them, and the only thing that can contribute to the constant number are terms with no variables on them. Thus, we can split the above into two equations:
 4x = Ax + Bx               −3 = A−6B
• While we can't solve for any numbers yet, we can put A in terms of B: A = 6B−3. We can then plug that into the other equation.
 4x = (6B−3)x + Bx,
which we can then solve to find B=1.
• Now that we know B=1, we can easily find A:
 A = 6 (1)−3     ⇒     A = 3
• Now plug our values of A=3 and B=1 into the decomposition we set up near the beginning to find how the fraction breaks down:
 4x−3 (x−6)(x+1) = A x−6 + B x+1 = 3 x−6 + 1 x+1
[3/(x−6)] + [1/(x+1)]
Find the partial fraction decomposition.
 11−7x x2+2x−35
• To decompose a fraction, the fraction must be proper. For this problem, we see that the numerator has a smaller degree than the denominator, so we're fine to continue. Next, we need to factor the denominator completely. This hasn't been done, so we factor x2+2x−35, allowing us to re-write the fraction as
 11−7x x2+2x−35 = 11−7x (x+7)(x−5)
• Once the denominator is factored, we can create an equation where the left-hand side is the original fraction and the right-hand side uses the denominator's factors and currently unknown constants. [See the lesson if you're not sure how to do this; it is fully explained in the lesson and much easier to understand visually.]
 11−7x (x+7)(x−5) = A x+7 + B x−5
• To solve for the constants on top of the fractions, multiply both sides by the factors in our starting fraction's denominator:
 (x+7)(x−5) · ⎡⎣ 11−7x (x+7)(x−5) = A x+7 + B x−5 ⎤⎦

 11−7x = A(x−5) + B(x+7)

 11−7x = Ax + Bx−5A+7B
• From here, we can solve based on each type. The only things that can contribute to the number of x's are terms with an x on them, and the only thing that can contribute to the constant number are terms with no variables on them. Thus, we can split the above into two equations:
 −7x = Ax + Bx               11 = −5A+7B

 −7 = A+B               11 = −5A + 7B
• While we can't solve for any numbers yet, we can put B in terms of A: B = −7−A. We can then plug that into the other equation.
 11 = −5A + 7(−7−A),
which we can then solve to find A=−5.
• Now that we know A=−5, we can easily find B:
 B = −7−(−5)     ⇒     B = −2
• Now plug our values of A=−5 and B=−2 into the decomposition we set up near the beginning to find how the fraction breaks down:
 11−7x (x+7)(x−5) = A x+7 + B x−5 = −5 x+7 + −2 x−5
[(−5)/(x+7)] + [(−2)/(x−5)]
Write the form of the partial fraction decomposition, but do not solve for the constants.
 x2−7 (x2+1)(x+7)2
• To decompose a fraction, the fraction must be proper. For this problem, we see that the numerator has a smaller degree than the denominator, so we're fine to continue. Next, we need to factor the denominator completely. This has already been done for us. [Notice that (x2+1) cannot be factored further because it is irreducible, so the denominator has been factored completely.]
• Once the denominator is factored, we can create an equation where the left-hand side is the original fraction and the right-hand side uses the denominator's factors and currently unknown constants. [See the lesson if you're not sure how to do this; it is fully explained in the lesson and much easier to understand visually.] Remember, if a factor is an irreducible quadratic (like x2+1), instead of a single constant on the top, it gets Ax+B. Also remember, if there are multiple of a single factor (like (x+7)2), a fraction appears for each multiple, and at each "exponent level".
 x2−7 (x2+1)(x+7)2 = Ax+B x2+1 + C x+7 + D (x+7)2
[(x2−7)/((x2+1)(x+7)2)] = [(Ax+B)/(x2+1)] + [C/(x+7)] + [D/((x+7)2)]
Write the form of the partial fraction decomposition, but do not solve for the constants.
 4x6+8x3−27x2+10 (x+2)(x−1)3(3x2−5x+4)2
• To decompose a fraction, the fraction must be proper. It would be difficult to expand the denominator, but we can see that its degree must be greater than the numerator's if we look at how much each factor would contribute to the overall degree. Next, we need to factor the denominator completely. It looks like this has already been done for us, but we should check to make sure that (3x2−5x+4) is actually irreducible. We can do so with the discriminant:
 b2−4ac     =     (−5)2 − 4(3)(4)     =     −23    < 0,
thus it is indeed irreducible, so the denominator is completely factored.
• Once the denominator is factored, we can create an equation where the left-hand side is the original fraction and the right-hand side uses the denominator's factors and currently unknown constants. [See the lesson if you're not sure how to do this; it is fully explained in the lesson and much easier to understand visually.] Remember, if a factor is an irreducible quadratic (like 3x2−5x+4), instead of a single constant on the top, it gets Ax+B. Also remember, if there are multiple of a single factor, a fraction appears for each multiple, and at each "exponent level".
 4x6+8x3−27x2+10 (x+2)(x−1)3(3x2−5x+4)2 = A x+2 + B x−1 + C (x−1)2 + D (x−1)3 + Ex+F 3x2−5x+4 + Gx+H (3x2−5x+4)2
[(4x6+8x3−27x2+10)/((x+2)(x−1)3(3x2−5x+4)2)] = [A/(x+2)] + [B/(x−1)] + [C/((x−1)2)] + [D/((x−1)3)] + [(Ex+F)/(3x2−5x+4)] + [(Gx+H)/((3x2−5x+4)2)]
Find the partial fraction decomposition.
 −7x−21 (x−5)(x2+3)
• To decompose a fraction, the fraction must be proper. For this problem, we see that the numerator has a smaller degree than the denominator, so we're fine to continue. Next, we need to factor the denominator completely. This has already been done for us. [Notice that (x2+3) is an irreducible quadratic.]
• Now we can follow the rules of decomposition to set up the equation we'll work with:
 −7x−21 (x−5)(x2+3) = A x−5 + Bx+C x2+3
• To solve for the constants on top of the fractions, multiply both sides by the factors in our starting fraction's denominator:
 (x−5)(x2+3) · ⎡⎣ −7x−21 (x−5)(x2+3) = A x−5 + Bx+C x2+3 ⎤⎦

 −7x−21 = A(x2+3) + (Bx+C)(x−5)

 −7x−21 = Ax2 + Bx2 −5Bx+Cx+ 3A−5C
• From here, we can solve based on each type. We effectively have three different types: x2, x, and constant. Thus, we can break the above equation into three separate equations:
 0 = A + B               −7 = −5B + C               −21 = 3A −5C
• We will solve through substitution. From the above, we can quickly see that B = −A. Plugging that in, we have
 −7 = −5 (−A) + C     ⇒     C = −5A − 7.
We can then plug that in to the last remaining equation to get
 −21 = 3A −5(−5A−7)     ⇒     −21 = 28A +35     ⇒     A = −2
• Now that we know A=−2, we can work out the others:
 B = −(−2)     ⇒     B = 2

 C = −5(−2)−7     ⇒     C = 3
• Now plug our values of A=−2, B=2, and C=3 into the decomposition we set up near the beginning to find how the fraction breaks down:
 −7x−21 (x−5)(x2+3) = A x−5 + Bx+C x2+3 = −2 x−5 + 2x+3 x2+3
[(−2)/(x−5)] + [(2x+3)/(x2+3)]
Find the partial fraction decomposition.
 3x3−4x2+16x−27 x4+10x2+25
• To decompose a fraction, the fraction must be proper. For this problem, we see that the numerator has a smaller degree than the denominator, so we're fine to continue. Next, we need to factor the denominator completely. This hasn't been done, so we factor x4+10x2+25, allowing us to re-write the fraction as
 3x3−4x2+16x−27 x4+10x2+25 = 3x3−4x2+16x−27 (x2+5)2
[Notice that (x2+5) is irreducible, so we're done factoring.]
• Now we can follow the rules of decomposition to set up the equation we'll work with:
 3x3−4x2+16x−27 (x2+5)2 = Ax+B x2+5 + Cx+D (x2+5)2
• To solve for the constants on top of the fractions, multiply both sides by the factors in our starting fraction's denominator:
 (x2+5)2 · ⎡⎣ 3x3−4x2+16x−27 (x2+5)2 = Ax+B x2+5 + Cx+D (x2+5)2 ⎤⎦

 3x3−4x2+16x−27 = (Ax+B)(x2+5) + (Cx+D)

 3x3−4x2+16x−27 = Ax3 + Bx2 + 5Ax + Cx+5B+ D
• From here, we can solve based on each type. We effectively have four different types: x3, x2, x, and constant. Thus, we can break the above equation into four separate equations:
 3 = A               −4 = B               16 = 5A + C               −27 = 5B + D
• We will finish solving through substitution. Plug in, using what we found above:
 16 = 5(3) + C     ⇒     C = 1.

 −27 = 5(−4)+D     ⇒     D=−7
• Now plug our values of A=3, B=−4, C=1, and D=−7 into the decomposition we set up near the beginning to find how the fraction breaks down:
 3x3−4x2+16x−27 (x2+5)2 = Ax+B x2+5 + Cx+D (x2+5)2 = 3x−4 x2+5 + x−7 (x2+5)2
[(3x−4)/(x2+5)] + [(x−7)/((x2+5)2)]
Find the partial fraction decomposition.
 3x3−7x2−32x+9 x2−3x−10
• To decompose a fraction, the fraction must be proper. However, for this problem, the fraction is improper because the numerator's degree (3) is greater than the denominator's degree (2). We can break the fraction into a non-fraction and proper fraction by using polynomial long division (if you're unfamiliar with polynomial long division, check out the lesson on it):
 x2
 −3x
 −10
 ⎞⎠
 3x3
 −7x2
 −32x
 +9

•  3x
 +2
 R:  (4x+29)
 x2
 −3x
 −10
 ⎞⎠
 3x3
 −7x2
 −32x
 +9
 3x3
 −9x2
 −30x
 2x2
 −2x
 + 9
 2x2
 −6x
 −20
 4x
 +29
Thus, by long division, we have shown that
 3x3−7x2−32x+9 x2−3x−10 =     3x + 2 + 4x+29 x2−3x−10
• At this point, we can now focus on decomposing the proper fraction from what we just figured out. We need to make sure we don't forget about the 3x+2 in front of the fraction we're decomposing, so make a note to put it back in at the end in front of whatever decomposition we figure out.
• Now that we have a proper fraction, we need to factor the denominator completely. This hasn't been done yet, so we factor x2−3x−10, allowing us to re-write the fraction we're working with as
 4x+29 x2−3x−10 = 4x+29 (x+2)(x−5)
• Once the denominator is factored, we can create an equation where the left-hand side is the fraction and the right-hand side uses the denominator's factors and currently unknown constants. [See the lesson if you're not sure how to do this; it is fully explained in the lesson and much easier to understand visually.]
 4x+29 (x+2)(x−5) = A x+2 + B x−5
• To solve for the constants on top of the fractions, multiply both sides by the factors in our fraction's denominator:
 (x+2)(x−5) · ⎡⎣ 4x+29 (x+2)(x−5) = A x+2 + B x−5 ⎤⎦

 4x+29 = A(x−5) + B(x+2)

 4x+29 = Ax + Bx−5A+2B
• From here, we can split the above into two equations (based on whether or not there's a variable attached), and work to solve from there:
 4 = A + B               29 = −5A+2B
• Use substitution to solve. We find that B = 4−A, so plug that in:
 29 = −5A + 2(4−A)
which we can then solve to find A=−3.
• Now that we know A=−3, we can easily find B:
 B = 4−(−3)    ⇒     B = 7
• Now plug our values of A=−3 and B=7 into the decomposition we set up to find how the fraction breaks down:
 4x+29 (x+2)(x−5) = A x+2 + B x−5 = −3 x+2 + 7 x−5
• Finally, don't forget that we started with a different fraction that we used long division on. We started with
 3x3−7x2−32x+9 x2−3x−10 =     3x + 2 + 4x+29 x2−3x−10 ,
and now, through the decomposition we just did, we've shown that
 3x3−7x2−32x+9 x2−3x−10 =     3x+2 + −3 x+2 + 7 x−5
3x+2 +[(−3)/(x+2)] + [7/(x−5)]

*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.

### Partial Fractions

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: 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
• Multiple Linear Factors
• Mixing Factor Types 10:28
• Figuring Out the Numerator 11:10
• How to Solve for the Constants
• Quick Example
• Example 1 14:29
• Example 2 18:35
• Example 3 20:33
• Example 4 28:51

### Transcription: Partial Fractions

Hi--welcome back to Educator.com.0000

Today, we are going to talk about partial fractions.0002

Consider if we had two quotients of polynomials that we wanted to add together, like, say, 7/(x - 5) and 3/(x + 2).0005

We could put them over a common denominator, then combine them; we have x - 5 here and x + 2 here,0014

so we can multiply the one on the left by x + 2, so we get (x - 5)(x + 2) on the bottom.0019

Its top will get also multiplied by that (x + 2).0026

For the other one, we will have (x - 5) multiplied on it, so it has (x - 5) multiplied on the top,0030

so we have (x - 5)(x + 2)--a common denominator--on the bottom now.0035

We multiply this out; we get 7x + 14 and 3x - 15; we combine those, and we get 10x - 1 on top.0039

divided by (x - 5)(x + 2), expanded to x2 - 3x - 10.0049

So, it is not a bad idea; we can get from here to here by putting it over a common denominator, and then just adding things out and simplifying.0054

But if we wanted to do the reverse process--what if we started with a fraction involving large polynomials,0064

and we wanted to break it into smaller fractions made of the polynomials' factors?0070

If we wanted to do this process in reverse, if we started at (10x - 1)/(x2 - 3x - 10),0074

and we somehow wanted to be able to get that into 7/(x - 5) and 3/(x + 2), what process could we go through?0080

We call the smaller fractions on the right partial fractions, because they are parts of that larger fraction.0088

So, each of these here (7/(x - 5) and 3/(x + 2))--they are each called a partial fraction.0095

And the process to break it up is called partial fraction decomposition.0101

So, whatever this method is that we haven't explored yet, to get from that larger polynomial quotient0105

into these smaller partial fractions, is called partial fraction decomposition.0111

So, how do we do it? To understand this lesson--to understand how we do it--you will need some familiarity with solving systems of linear equations.0116

Previous experience from past algebra classes will probably be enough to get through this.0124

But if you want a refresher, if you are a little confused by how this stuff is working later on,0128

check out the lesson Systems of Linear Equations--that will help explain this stuff if it confuses you right now.0132

You also need to know how to factor polynomials, and you will need to understand them in general,0138

along with the ability to do polynomial division; you will need all of that for some of the stuff in here, as well.0141

Now, sadly, we won't be able to see the application of partial fraction decomposition in this course.0147

However, it is quite useful in calculus, where it will allow us to solve otherwise impossible problems.0152

Partial fraction decomposition is this really useful thing that lets us break up these complicated things0157

that we couldn't solve, and turn them into a form that is actually pretty easy to solve.0162

It is really handy in calculus.0166

We won't be able to see its use in this course; we won't see it any time soon.0167

But it is helpful to practice it now, just like we practiced stuff about factoring complicated polynomials in algebra,0171

before we really had a great understanding of what it was all about.0176

We are practicing something so that we can use it later on.0179

And also, it just is a great chance to flex our brain and get some mental "muscles" that are kind of difficult ideas, but really require some analytic thought.0182

All right, let's get to actually figuring out how to do this.0191

If we have a polynomial fraction in the form numerator polynomial divided by denominator polynomial,0194

a normal rational function format, there are two possibilities in regards to the degrees of the top and bottom polynomials.0199

We call them proper (proper is when the degree of the numerator polynomial is less than the degree of the denominator polynomial--0206

the degree of n is less than the degree of the denominator) and improper (when the degree of our numerator0214

is greater than or equal to the degree of the denominator).0222

To decompose the fraction, it must be proper; we have to be in this proper format.0226

We have to make sure that the degree of our numerator is less than the degree of our denominator, if we want to do partial fraction decomposition.0231

So, what do we do if the fraction is improper?0239

We have to turn it into a proper thing; so if the fraction is improper, and we want to decompose it,0241

we have to make it proper through polynomial division.0245

We can make that numerator smaller by being able to break it off and divide out the part that isn't smaller and the part that is smaller.0249

Remember, when you divide your polynomial division, the remainder from the division0256

goes back onto our original denominator, which we will then decompose.0261

We will be able to decompose the part that is the remainder.0265

The other part that comes out cleanly--well, that is just going to be there.0267

That is just going to be a polynomial that is then going to be added to whatever ends up coming out of our decomposition.0271

We will see an example of this in Example 3.0276

Once we have a proper polynomial fraction to decompose into partial fractions0280

(remember, proper: our numerator has to be smaller than our denominator before we can enact this process),0284

once we have done that, we can then factor the denominator.0289

Our next step is to factor the denominator.0293

After the denominator is broken into its smallest possible factors, we are ready to decompose.0295

Now, there are two types that the smallest possible factors can come in.0300

They can come in linear factors, which is forms ax + b, and they will be raised to some power, because we might have a multiplicity of these factors.0302

There might be multiple of a given factor, so if we have multiple ax + b's, we will have m of them, (ax + b)m.0309

So, we will have linear factors in this; or we could have irreducible quadratic factors,0316

that is, things ax2 + bx + c that can't be broken up further into linear factors.0320

There is no way to break them up further in the reals.0325

So, ax2 + bx + c...and once again, it will be raised to some power;0328

there will be m of them, so it is raised to the m, because there are m of them multiplying together.0331

Remember: "irreducible" means it can't be broken up further in the reals.0336

That means quadratics like x2 + 1 or 5x2 - 3x + 20, because they have no roots.0339

x2 + 1 has no roots; x2 + 1 = 0...there are no solutions, in the reals, at least.0346

If we allow for the complex, there are; but we are not working with the complex.0355

There are no solutions in the reals; so, since there are no solutions in the reals, it can't be factored any further.0358

x2 + 1 is irreducible, therefore.0364

The next step of decomposition will behave differently, depending on which flavor of factor we have--0366

whether we are at a linear factor or we are at an irreducible factor; and we will look at the two, one after another.0371

Linear factors: the partial fraction decomposition must include the following for each linear factor that is in this form (ax + b)m.0377

So, it is going to have this in its decomposition: A1/(ax + b) + A2/(ax + b)2 +...0385

notice that we are doing this where it is going to keep stepping up with this exponent on the bottom,0394

as we keep putting these things in, until eventually we get up to our mth step,0398

because we had m of them to begin with, so we have to have m of them in the end.0402

And each one of these A's on top (A1 up until Am) are all just constant real numbers.0406

We are using A with a subscript, this A1, A2, A3 business,0411

because we just need a way of being able to say m of them, and we are not quite sure that we are going to go only out to m.0415

Anyway, the point is that they are all going to be constant real numbers.0420

For example, if we had (x + 7)3, then we have some stuff up top.0423

We don't really care what the stuff is right now, because we just want to see how it will break out.0428

We will deal with the stuff later.0431

If we have stuff over (x + 7)3, then it is going to break into 3 of these, because of this "cubed."0433

We will have A/(x + 7) + B (we can switch into just using letters in general, because we know0439

that each one of these capital letters just means some constant real number; we will figure them out later--0444

that will come up later--don't worry)...A/(x + 7) + B/(x + 7)2 + C/(x + 7)3.0449

Notice: we started cubed, and so we have three of these; we step up each time with the exponent,0457

until we have gotten to the number of our multiplicity that we originally started with--0462

whatever the exponent was originally on the factor.0465

If we have multiple linear factors, we do it for each one of these; so if we have (2x + 3)2, x10469

(since there is nothing there), (x - 5)1 (since there is nothing there)...we have A/(2x + 3) + B/(2x + 3)20475

(because remember: it was squared to begin with, so it has to have 2 of itself) + C/x.0483

And there is only to the 1, so there is only going to be one of it; plus D/(x - 5).0490

And it is only 1, because there is only one of them to begin with.0495

And each one of these A, B, C, D...they are each just constant numbers.0498

We are going to figure them out later on; don't worry.0503

Irreducible quadratic factors: the method is very similar for irreducible quadratic factors.0507

The decomposition must include the following for each irreducible quadratic: ax2 + bx + c to the m.0511

Once again, this is the same thing of stepping out m times.0516

A1x + B1 over ax2 + bx + c: notice how, previously, it was A over some x plus a constant,0520

because we had that the degree of the top was one below the degree on the bottom.0528

So, the degree on the top here is a linear factor on top, because we have a quadratic factor on the bottom.0532

Previously, we had a constant factor on top, because we had a linear factor on the bottom.0537

So, A1x + B1 + A2x + B2, over...that should be a capital...0541

ax2 + bx + c, now squared, because we are doing our second step...0548

we do this out m steps, until we are at our m constants, and we are at the mth exponent on the bottom.0552

So, A1...all of these capital letters...B1 to Bm...they are all just constants.0558

And we will figure them out later.0564

For example, if we had (x2 - 4x + 5)2 on our bottom, then we have some stuff--0565

it doesn't really matter right now--x2 - 4x + 5, squared; so we are going to have this happen twice.0571

The first one will be just to the one; the second time, it will be squared.0576

And on the top, it is going to be Ax + B and Cx + D.0580

So, A, B, C, D...these are all just constants; we will deal with them later.0585

If we have multiple irreducible quadratics, we just do each of these.0590

For example, if we have x2 + 2x + 3, that is an irreducible; and x2 + 4, squared...0593

then we have Ax + B, x2 + 2x + 3...there is only one of them, so it only shows up once,0599

+ Cx + D over x2 + 4...it is going to show up twice because of that 2, so it shows up a second time here.0605

And we have Ex + F; so we just keep doing this process with capital letter, x, + capital letter, plus capital letter, x, plus capital letter,0612

until we have worked out all of the times that we had things showing up--a total of 3, because we had one there and two there.0620

So, we have a total of 3.0627

If we have both types, a linear and an irreducible quadratic, mixed together in the denominator, that is OK--perfectly fine.0629

We just decompose based on both of the rules.0634

So, for example, if we had (x + 9)2, then the x + 9's are going to follow this A, this single-constant format,0637

because linears just had one constant on the top; x + 9, and then (x + 9)2.0645

And then, we switch to the other rule, x2 + 1; now we are dealing with an irreducible quadratic.0650

So, x2 + 1...and we have Cx + D on top, because we have to be able to have...0656

since we have a quadratic on the bottom, we now have linear factors on the top.0661

When we are dealing with linear factors on the bottom, even if it is multiple linear factors, it is just constants on the top.0665

All right, now, finally: how do we figure out what those numerators are?0671

We have mentioned that the numerators are built out of constants.0674

Remember: A for linear factors (constants for linear factors); Bx + C...linear factors for irreducible quadratic factors.0677

But we haven't talked about how we actually find out what these capital letters' values are.0685

We solve for the constants by doing partial fraction decomposition.0690

We set them up in this format, and then we multiply each side of the equation by the original denominator.0693

It will make more sense as we work through examples; let's look at this example.0698

If we have (3x2 + 3x - 4)/(x + 3), one factor, times (x2 - x + 2), an irreducible quadratic factor,0701

then we would be able to break it into A/(x + 3) (we get this right here)...and x2 - x + 2 would be Bx + C/(x2 - x + 2).0709

So, we know, from what we just talked about (this process that we just went through) that that is how it breaks up.0721

But how do we figure out what A and Bx + C are?0726

Well, notice: we can just work things out the way we would a normal algebra thing.0728

We multiply both sides, because we want to get rid of this denominator.0734

We will multiply by x + 3 on the left side, and x2 - x + 2 on both sides.0737

So, we do the same thing on this side, as well: (x + 3)...we will color-code this...and (x2 - x + 2).0746

So, the (x + 3) will distribute to the one on the left and the one on the right.0759

It has to hit both of them, because it is distributed, because there is this plus sign right here.0764

So, (x + 3) distributes to both of them; similarly, (x2 - x + 2) will distribute to both of these, as well.0768

Now, notice: on the left side, we have (x + 3) on the bottom, and we are multiplying by (x + 3); so the (x + 3)'s cancel out.0776

(x2 - x + 2) is right here and here; they cancel out.0783

What about on the left side? Well, we have (x + 3) and (x + 3); so here, it is going to cancel this and this.0788

But it will still have the (x2 - x + 2) coming through.0796

What about the Bx + C over (x2 - x + 2)?0799

Well, the (x + 3) is still going to come through--it doesn't cancel out any factors here.0801

But (x2 - x + 2) is the same thing, so it will cancel out that part of the factor coming in.0805

So, we will see that we have A times (x2 - x + 2) and Bx + C times (x + 3) coming through.0809

This part right here manages to come through on the Bx + C, and this part right here manages to come through on the A.0819

And on the left, since we canceled out everything on the bottom, we just have what we had on our numerator originally.0828

That just makes its way down; so we have 3x2 + 3x - 4 = A times x2 - x + 2, plus Bx + C times (x + 3).0834

Which...we can then expand everything on the right, and we can work things out by creating a system of linear equations.0845

The examples are going to help greatly in clarifying how this works, so check them out,0851

because we will actually see how this process of setting up our system of linear equations works,0855

and solving for what these values are, in each of these examples.0859

Not Example 2, but we will see it in a lot of the examples.0863

And it will make a whole lot more sense as we see it in practice.0866

So, let's go check it out: the first example is a nice, simple example to get things started with.0868

Find the partial fraction decomposition for (4x + 3)/x(x + 3).0873

So, we have 4x + 3, over x(x + 3); that is going to be equal to...we have two linear factors on the bottom, so A/x, plus our other linear factor, B/(x + 3).0878

Now, at this point, we can multiply everything by x and (x + 3).0900

Great; so, on the left side, that is just going to cancel this stuff out; and on the right side, it will end up distributing.0912

So, on the left side, we have 4x + 3 = A...now, notice: we have the x multiplying here; that is going to cancel out.0917

So, the x's parts get canceled out; but we are left with x + 3 multiplying on it.0926

Plus B times (x + 3)...so the x + 3 part cancels out; I will switch colors here.0931

x gets through; but the (x + 3) cancels out the denominator; great--we have managed to get rid of all of our denominators.0939

That is great; it will make things easier to see what is going on.0946

And we have some stuff multiplying through; so at this point, 4x + 3 on the left equals Ax + 3A + Bx.0949

Now, notice: we see here that we have a 3 constant.0965

Now, what constants do we have on the right side?0970

We have to have all of our x's match up and all of our constants match up.0972

Well, the only thing that is actually a constant on the right side is this 3A, because we have Ax and Bx.0976

But we don't have any B just as B; we don't have any constants of B.0982

So, it must be that 3 and 3A are equal, because they are all of the constants that show up on either side of our equation.0986

The same thing has to be on the left and the right; otherwise, it is not an equation.0992

That tells us that 3 = 3A; so what is our A?0996

We divide both sides by 3; we get 1 = A; so now we have figured out what A is.1001

What about the other part? Well, we have that 4x; 4x is equal to...let's give it a special thing; we will give it a curly around...1007

4x is equal to all of our x's on the right side, put together.1019

So, that is our Ax, combined with our Bx; so it must be the case that 4x is equal to Ax + Bx,1023

because the same number of x's must be on the right side as on the left side.1034

So, 4x must equal Ax + Bx, because these are our sources of x on the right side.1039

4x = Ax + Bx; we just figured out what A is, so we can plug that in over here; so we have 4x = Ax + Bx...1045

Now, notice...actually, before we do that, even, we have 4x = Ax + Bx; so if that is the case, let's just get rid of these x's.1055

All right, we know that it must be the case that, since 4x = Ax + Bx...well, that has to be true for any x that we plug in, at all.1063

So, it must be the case that 4 = A + B; we can divide the x out on both sides, and we cancel that to 4 = A + B.1072

Now, we can substitute in that 1; that will make it even clearer what is going on: 4 = 1 + B.1082

Subtract the 1; we get 3 = B; so at this point, we have figured out what A is; we have figured out what B is.1088

So, we see that we can now go back to our partial fraction decomposition.1094

We can plug in actual numbers, and we can have what that is equivalent to; and we have 1 over (x + 3) over (x + 3).1098

And there is our answer; great.1111

All right, the next one: Write out the form of the partial fraction decomposition, but do not solve for the constant.1114

That is good, because this would be really hard, if we actually had to solve it.1120

So, first we have x, and then (x - 7)2, and then 2x2 + 1, an irreducible quadratic, cubed.1123

All right, we have A/x, plus...the next one is a linear, so it is also just one constant up top, (x - 7);1131

but it is squared in our original thing, so it gets squared; so another constant...C/(x - 7)...this time,1142

it is two of them; it is squared; plus Dx + E, because now we are dealing with an irreducible quadratic.1149

The first time it showed up, so 2x2 + 1, plus Fx + G...just keep going with letters;1158

I am just counting off the alphabet at this point, Fx + G...A, B, C, D, E, F, G...I don't want to make a mistake with my alphabet!...1166

2x2 + 1...now we are at the second time, so squared...Hx + I, over (2x2 + 1)3.1173

And that is the partial fraction decomposition.1186

We haven't figured out what A, B, C, D, E, F, G, H, I are.1187

But we could, if we multiplied, because we know that this thing here is equal to this thing here, once we get those correct constants in.1193

So, we could multiply the left side and the right side by that denominator.1202

And everything would cancel out, and we would be left with this awful, awful massive thing.1206

But we could solve it out, slowly but surely; luckily, all it asked for was to just set things up--1209

set up the partial fraction decomposition, but not solve for these constants.1215

So, happily, we can just see that this is how this breaks down.1219

We break it down this way; we use each of the rules, based on whether it is linear or it is an irreducible quadratic.1222

And they show up the number of times of each of their exponents.1227

Great; the third example: Find the partial fraction decomposition for (x5 + 3x3 + 7x)/(x4 + 4x2 + 4).1232

The first thing to notice is that it has a degree 5 on top and a degree 4 on the bottom, so we start as improper.1241

So, since it is improper, we have to use polynomial division to break it into a format that is proper, first.1248

How many times does x4 + 4x2 + 4 go into this thing here?1254

So, let's set up our polynomial division: 4x2 + 4 goes into x5 +...we have no x4's...1258

plus 3x3...we have no x2's...+ 7x; and we have no constants.1270

OK, how many times does x4 go into x5?1279

It goes in x, because x times x4 gets us x5; and we do it on the other things...+ 4x3 + 4x.1281

So, we subtract all of this; let's distribute that subtraction...1294

x5 - x5 becomes 0--no surprise there.1299

3x3 - 4x3 becomes -x3; 7x - 4x becomes positive 3x.1303

We can now bring down everything else, but as soon as we do that, we realize that there is nothing else to be done here,1311

because how many times does x4 go into 0x4? It goes in 0 times.1324

It can't fit in at all, because it is just a 0 to begin with.1329

So, we found our remainder; our remainder is what remains: our -x3 here, our 3x, and that is it.1331

-x3 + 3x is what remains.1338

That means that what we originally started with, x5 + 3x3 + 7x,1342

all over x4 + 4x2 + 4, is equal to...x came out of it, so x plus what our remainder was,1350

-x3 + 3x, over that original denominator, x4 + 4x2 + 4; great.1366

We need some more room; let's flip to the next page.1377

That is what we figured out: we figured out that what we started with breaks into this thing right here.1380

So, now let's figure out how we can decompose this part on the right.1386

We will work on decomposing this first, but we can't forget (circle it in red, so we don't forget) that x, because that is still part of that function.1391

We can't actually decompose it without putting that back in at the very end.1400

Otherwise, we will have decomposed a different function; we will have decomposed this function right here,1403

but forgotten about what we originally started with, because what we originally started with includes this x.1408

All right, -x3 + 3x...oh, that is one thing I forgot to do last time.1413

We had x4 + 4x2 + 4; I quietly factored that, without ever mentioning that it factors into (x2 + 2)2.1418

I'm sorry about that; hopefully that isn't too confusing.1434

-x3 + 3x...over (x2 + 2)2...that is going to be equal to Ax + B,1437

because it is an irreducible quadratic, divided by x2 + 2, plus Cx + D, over (x2 + 2)2.1446

At this point, we multiply both sides by (x2 + 2)2, (x2 + 2)2, (x2 + 2)2.1458

So, that cancels out here, and we are left with -x3 + 3x =...what happens to the Ax + B?1470

Well, its denominator gets canceled, but then it is still left getting hit with one more x2 + 2.1478

What about Cx + D? Well, its denominator just gets canceled; there is nothing left after (x2 + 2)2 hits,1485

because its denominator was the same thing, so we are just left with Cx + D.1492

So, let's expand that, and let's also put this in a different color, just so we don't get confused by the separation.1495

I am not sure how much that will help us.1503

3x =...Ax times x2 becomes Ax3; Ax times 2 becomes 2Ax.1507

Bx2 + 2B...and we will also bring along + Cx + D.1515

At this point, we can set things up so we can see it a little more clearly by putting everything together that comes in a single form.1524

We have Ax3, and we will leave a blank for the x2's...+ 2Ax.1534

And then, we can write over here + Bx2 + 2B...oops, sorry; that is going to have to leave a blank there, as well...1541

+ 2B...and then + Cx, and then + D; notice how this works.1550

We have our cubes here (degree 3 things here); degree 2 things here...1559

well, not quite "degree," because they are not the whole polynomial,1566

but things with exponent 3 here, exponent 2 here, exponent 1 here, and exponent 0 here.1568

So, now we can compare that over here, where we have exponent 3 here and exponent 1 here.1573

So, all of the things that are going to connect to the -x3, which we can also see as -1 times x3, are just Ax3 here.1578

So, it must be that -1 is equal to A; otherwise, we wouldn't have -x3 in the end.1587

We now know that -1 = A; great.1593

What about 2Bx2? Well, we have 0x2; we know that 0x2 = Bx2.1598

Things of degree 2...well, we could have just written that as 0 = B, because our things of degree 2 have to line up with that number B.1608

So, 0 = B; what about our D here? We can figure out our D.1615

Well, we have a constant...+ 0 here; so it must be that 0 is equal to 2B + D.1622

Well, we already know that B equals 0 here; so that just knocks out; so we are now told that 0 equals D, as well.1632

Finally, we are left figuring out what C is; so we know that 3, here, equals 2A, because there are that many x,1641

because we have 3x and 2Ax and Cx, so it must be that 3 = 2A + C, when we combine them all together.1650

We plug in numbers: 3 = 2 times...what is our A? It is -1, so -1, plus C; so we get 3 = -2 + C, or we add 2 to both sides: 5 = C.1662

All right, so 5 = C; so at this point, we have managed to figure out everything that goes into our decomposition.1684

So, we can write that decomposition out: Ax + B...we have -1 as our A, so -x...what is our B?1692

Our B is 0, so it just isn't anything; x2 + 2, plus Cx + D...our D was 0; our C was 5.1701

So, it is going to be 5x/(x2 + 2)2.1712

And we can't forget that x that was originally there; so x plus that...that is our entire partial fraction decomposition.1719

It breaks down into that one right there.1728

All right, the final example: this one is a rough one, but this is the absolute hardest you would end up coming across in class and a test.1731

So, don't worry: this is basically the top difficulty you will have to deal with, probably.1741

You might see something a little bit harder, but this is really about as hard as it is going to get.1746

All right, so happily, they already factored it for us, and we have (t - 2)2(t2 + 2).1750

So, this is degree 2; this is also degree 2; so that combines to degree 4 on the bottom, degree 3 on the top.1756

So, we are proper; our bottom is already factored; we just need to actually work through the decomposition now.1762

We have -t3 + 8t2 - 6t, over (t - 2)2(t2 + 2).1769

How does that decompose? A over our linear factor, t - 2, plus B over our second version of that linear factor,1781

(t - 2)2, plus Cx + D over our irreducible quadratic of t2 + 2; great.1792

At this point, we multiply both sides by the denominator, and we will get (-t3 + 8t2 - 6t) =1801

A times...it was hit with (t - 2)2 + (t2 + 2), so it is going to cancel out one of those (t - 2)'s,1814

and it will be left being multiplied by (t - 2) and (t2 + 2).1822

Plus...B: (t - 2)2 was in its denominator, so it is going to cancel out all of the (t - 2)2 that hit it, but not the t2 + 2 at all.1828

So, we have t2 + 2.1838

And then finally, plus Cx + D, and it is going to have to go in parentheses, because it was hit by another entire thing.1842

And it is going to get (t - 2)2, but it did cancel out the (t2 + 2) that hit it, because it had that in its denominator.1849

At this point, we want to start simplifying things out.1856

We have A times...what is t - 2 times t2 + 2?1859

We get t3...t times 2 becomes 2t...minus 2t2, minus 4,1863

plus Bt2 + 2B + (Cx + D) times (t - 2)2, which becomes t2 - 4t + 4.1877

All right, move a little bit to the left, so we have more room; it is still the same thing on the left side.1893

We have At3 - 2At2 + 2At - 4A (oops, sorry--I just switched colors there)1899

plus Bt2 + 2B...break onto a new line now, so I can see what is going on...1920

Cx times t2 will become C...oops, I made a mistake right from the beginning.1926

It should not be x; it should be (I just stuck with what I have been used to)...the variable we are dealing with isn't x; it is t here.1932

So, that should have been a t the whole time; sorry about that, but that is the sort of mistake you want to catch on your end, too,1942

because we are dealing with t as our variable.1951

So, while the form was with x, as before, that is because the variable was x before.1955

But now, we are dealing with t as a variable, so the form needs to switch to using t.1958

So, let's work this out: Ct times t2 becomes Ct3; Ct times -4t becomes -4Ct2;1963

Ct times +4 is + 4Ct; + Dt2 - 4Dt + 4D; OK, we have a lot of stuff here.1973

So, we can say that this is (-t3 + 8t2 - 6t) =1985

and we can put this in that form, again, of thing with exponent 3, exponent 2, exponent 1, exponent constant;1993

minus 2At2 + 2At - 4A; the next one is + Bt2 + 2B + Ct3 - 4Ct2 + 4Ct,2000

and then + Dt2 - 4Dt + 4D.2027

So, from this, we will be able to figure out that all of our t3's, our line of t3's here,2036

has to be the same as our line of t3's here.2042

So, we will be able to get things like -1 = A + C.2044

Now, this is a lot of stuff to have to work through; we have 4 different variables;2048

we are going to end up having different simultaneous linear equations that we are going to have to solve through.2053

If you have done much work with simultaneous linear equations, you know that that is going to be kind of a pain to work through.2057

So, at this point, we might think, "Oh, I'm lazy; is there anything clever I can do?"2062

Is there a clever way to work through this--some little trick I could use?2067

Well, if we look back to what we originally had when we multiplied out that denominator,2071

we might think, "Oh, look, there is a t - 2 here; there is a t - 2 here; but there is an absence of t - 2 on our B."2076

So, if we realize that, we could say, "Well, if we plug in t = 2, that would cause..."2086

now, none of this stuff up here is going to be true; but we are going to plug in t = 2;2093

and we are going to realize that, if we plug in t = 2, that causes this to turn to 0, which knocks out our A terms.2099

And it will cause this here to turn to 0, as well, and also knock out our C and our D terms.2105

And we will be left with just B times t2 + 2, where t is 2--only t is 2--2110

and -t3 + 8t2 - 6t, when t is 2.2117

Now, this is true for any t; so that means we can plug in any t we want,2122

and if there is a convenient way to get certain things to disappear by plugging in a cleverly chosen t, it is fair.2127

Anything we can do to help us work through this--anything that makes it easier--is fair.2134

So, in this case, we notice that by plugging in this carefully-chosen t, we can get certain things to disappear.2138

Now, it is true for any t; this equation, this whole thing here, is true in general for any t, before we cross stuff out.2144

But in the specific case where we plug in t = 2, certain things will cancel out, and we will be able to figure things out in a very easy way mathematically.2152

So, we plug this in; and we know that, on our left side, we are going to have -23 + 8(2)2 - 6(2).2161

That equals (sorry, there is not quite a lot of room here; I am going to draw in a line, just so we don't get confused)...2173

That equals B times 22 + 2; OK.2180

So, we have -22 is 8, plus 8(2)2 is 4, so 32, minus 6 times 2 is 12, so - 12, equals B times 4 + 2, or 6.2186

Simplify the left side: -8 - 12 is -20; 32 - 20 gets us 12, equals 6B; we divide both sides by 6, and we get 2 = B.2208

So, by use of some cleverness, we were able to skip having to do four simultaneous linear equations,2221

to where eventually we are effectively going to bring it down to 3.2227

And we will be able to not have to get confused by this most complicated column, where we have four different things all going together.2230

We can just be done with B; B is already figured out, which will be really helpful and make things easier on us later on.2236

We could have solved out those four simultaneous linear equations, because each one will produce things.2242

Remember how the first column produced -1 = A + C; each one of these columns will produce an equation.2247

And from four simultaneous linear equations, solving for four variables, we will be able to do it; it is just kind of a pain.2253

So, we came up with this clever way, and we were able to figure out 2 = B by just happenstance.2259

If we plug in t = 2, it made everything but the B terms disappear, and that just left us with a single equation that was pretty easy to solve.2263

And we figured out that 2 = B; all right, well, let's work through it now.2271

So, we figured out B = 2, and now we have -t3 + 8t2 - 6t =2275

At3 + Ct3 - 2At2 + Bt2 - 4Ct2 + Dt2 + 2At + 4Ct - 4Dt - 4A + 2B + 4D; wow!2282

OK, so at this point, let's put columns to columns; so -t3 goes to our At3 + Ct3.2294

So, it must be that -1 is equal to A + C.2301

We can do this column here; I accidentally cut through that negative sign--hopefully that wasn't too confusing.2309

So, that one here lines up with the -6t; so we have that -6 (let's write it on a separate location) equals 2A + 4C - 4D.2316

They all have t's showing up, so we just divide out the t's, and we are left with this simple linear equation that we can work with.2334

And then, finally, the last one of constants: what is our constant? Our constant is 0.2340

So, we have 0 = -4A + 2B + 4D; for extra credit, let's just see what this would have been.2347

How many t2's did we have? We had 8, so in our difficult-to-read yellow (I'll make it black,2357

just because black is easier to read), 8 = -2A + B - 4C + D.2362

We could have worked that one out, but I wanted you to see what it would have been.2371

But we will actually end up having enough information, because we have three equations and three unknowns;2375

so we have enough with the red, blue, and green things.2379

So, at this point, let's figure out A first; so that means we need to solve for everything that isn't A,2382

so that we can plug them in, get rid of everything else, and just have A.2389

So, on the left side, we have -1 = A + C; so we solve for C, and we get -A - 1 = C.2392

OK; on the green one, we plug in what we know for our B.2405

We have 0 = -4A + 2(2 + 4D); 0 = -4A + 4 + 4D; 0 =...let's divide everything by 4, just to make things easier...-A + 1 + D.2409

So, we move that stuff over, and we get: add A on both sides; subtract by 1 on both sides; A - 1 = D.2430

So, we have A - 1 = D; we have -A - 1 = C; so we can plug this information into this equation, and we will be able to get something that is just using A.2439

-6 = 2(A)...that actually stays around, because it was just A from the beginning; 4...what did we figure out C was?2460

C is the same thing as -A - 1, minus 4 times...what did we figure out D is?2468

It is the same thing as A - 1; simplify that out: -6 = 2A + 4(-A), so we will make that - 4A, minus 4...2475

-4A - 4A...and -4 times -1 becomes + 4; so at this point, we see that we have -4 and +4, so those cancel each other out.2489

We have -6 = 2A - 8A; -6 = -6A; and thus 1 (dividing both sides by -6) = A.2500

With 1 = A in place, we can now easily go back and figure out everything else.2513

We have -A - 1, so negative...plug it in, knowing that it is 1...minus 1, equals C; so we have -2 = C; great.2518

And also, plug in over here; and we have that 1 - 1 = D; it turns out that D is just 0.2533

So, at this point, we have figured out all of the things that we need.2545

We have this set up as (t - 2) + (t - 2)2 + (t2 + 2).2549

It was in the format -t3 + 8t2 - 6t, all over (t - 2)2(t2 + 2).2557

OK, so that is the case; we have that as something over (t - 2).2579

It was A/(t - 2) + B/(t - 2)2 + (Cx + D)/(t2 + 2).2586

OK, at this point, we can actually plug in our numbers; we have that A is 1, so we have 1/(t - 2).2601

B is 2, so we have 2/(t - 2)2, plus...C was -2, so -2x, and 0 was D, so that part just disappears.2608

So, we have -2x/(t2 + 2); oops, once again, I did it again; I got to thinking in terms of x, as opposed to t.2624

But we are using a different variable: -2t/(t2 + 2).2633

And there we are; we have decomposed it into its partial fractions.2638

That is pretty long; it is pretty complicated; but that is pretty much as hard as it gets.2643

As long as you break it down, and then you multiply everything out carefully,2647

and you are careful with all of your algebra and your arithmetic, you can get it into this form right here,2651

where you have this big block compared to these original things, and then you are able to figure out all of these linear equations.2656

And you can possibly be clever and figure out some way to figure out B = 2.2662

But you also can just work through a bunch of linear equations, solving for one thing at a time in terms of the other,2667

plugging them all together, and then solving it out.2673

Sometimes it takes a lot of work; sometimes it goes kind of slowly.2675

But ultimately, if you just keep working at it, you can get it.2678

And it is really, really useful in calculus.2680

I know--I know that you aren't going to be using it immediately.2683

But honestly, this thing makes a problem that would be, otherwise, totally impossible to do, just really easy.2685

So, it is a really handy thing in calculus.2691

All right, we will see you at Educator.com later--goodbye!2693