Welcome to College Algebra.
G.H. Hardy, a famous 19th Century mathematician wrote:
"A mathematician, like a painter or poet, is a maker of patterns.
If his patterns are more permanent than theirs, it is because they are made with ideas..."
This will be a course with emphasis on IDEAS. An idea, like a picture, is worth 1000 words!
With the ideas clear, language & techniques become easier to remember;
they hang together in your mind.
This course will prepare you for further courses, such as
Analytic Geometry, Precalculus, and Calculus.
Please note: A graphing device is appropriate for this course, and will be discussed.
However, I expect YOU to master the basic features of your device yourself.
My recommendations to you:
Replay any tape segment whenever you feel it's necessary!
This is the great advantage of a course on video.
Pay attention to the language I use in speaking about Algebra.
Try to become fluent in this language yourself.
Adopt an organized style of work, as I do.
You need not mimic me exactly, but learn to leave a clear "paper trail",
especially when you use a graphing device.
Don't get behind! Work lots of problems.
Now it's time we roll up our sleeves and get down to the nuts-and-bolts of Algebra!
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UNIT 0 - BASICS: Remembrance of Things Past
[6 hr. 49 min.]
Tape 1
NUMBERS [1 hr. 14 min.]
- Sets of Objects [4 min.]
- Natural Numbers [7 min.]
- Integers [5 min.]
- Rational Numbers [13 min.]
- Irrational Numbers [2 min.]
- Real Numbers [1 min.]
- The Real Line: From Numbers to Points [4 min.]
- Real Numbers are Ordered [19 min,]
- The Real Line: Distance Between Points [8 min.]
- Real Numbers as Decimals [11 min.]
Tape 2
THE LANGUAGE OF MATHEMATICS [27 min.]
- Learning to Read Mathematics [7 min.]
- Some Symbols of Algebra [9 min.]
- Nouns, Pronouns, & The Main Verb of Algebra [4 min.]
- Theorems, Corollaries, Lemmas, and All That [7 min.]
Tape 3
THE POWERS THAT BE: EXPONENTS [1 hr. 29 min.]
- Integer Exponents [12 min.]
- Operations with Integer Exponents [31 min.]
- Square Roots: A Pair Of Equal Factors [13 min.]
- Nth Roots & Rational Exponents [12 min.]
- Operations with Rational Exponents [21 min.]
Tape 4a
POLYNOMIAL EXPRESSIONS [3 hr. 2 min.]
- What is a Polynomial? [18 min.]
- Adding & Subtracting Polynomials [3 min.]
- Multiplying Polynomials [19 min.]
- A Common Error [5 min.]
- Handy Polynomial Products [15 min.]
- Un-Multiplying (Factoring) Polynomials [28 min.]
- Completing A Perfect Square [10 min.]
Tape 4b
- Dividing Polynomials: Rational Expressions [24 min.]
- The Art of Simplification [14 min.]
- Solving Some Polynomial & Rational Equations [46 min.]
Tape 5
MORE NUMBERS [19 min.]
- Beyond Real Numbers: Complex Numbers [19 min.]
A LITTLE GEOMETRY [20 min.]
- Some Area Formulas [8 min.]
- The Pythagorean Theorem & A Visual Proof [12 min.]
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UNIT 1 - GRAPHS
[3 hr. 53 min.]
Tape 6
GRAPHS [3 hr. 53 min.]
- Rectangular Coordinates: Geometry Meets Algebra [11 min.]
- Distance Between Points [18 min.]
- Midpoint of a Line Segment [7 min.]
- Your Graphing Device [17 min.]
Tape 7
- Graphs of Equations [25 min.]
- Intercepts: Crossing the Axes [12 min.]
- Symmetry of Graphs [14 min.]
- Lines: Defining "Slope" [34 min.]
Tape 8
- Lines & Their Equations [24 min.]
- Parallel & Perpendicular Lines [26 min.]
- Circles & Their Equations [28 min.]
- Some Exercises Explained [16 min.]
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UNIT 2 - FUNCTIONS AND THEIR GRAPHS
[7 hr. 2 min.]
Tape 9
FUNCTIONS AND THEIR GRAPHS [7 hr. 2 min]
- Function: The Central Idea of Mathematics [26 min.]
- The Language & Notation of Functions [20 min.]
- More On Domains [19 min.]
- Function Notation Practice [16 min.]
Tape 10
- Visualizing Functions: Graphs of (x, f(x)) Pairs [26 min.]
- Increasing & Decreasing Functions [23 min.]
- Local Maximums & Local Minimums [18 min.]
- Even & Odd Functions [20 min.]
Tape 11
- A Library of Important Functions [20 min.]
- Piecewise Defined Functions [19 min.]
- Some Exercises Explained [11 min.]
Tape 12
- Graphing Techniques: Vertical & Horizontal Shifts [28 min.]
- Graphing Techniques: Compressions & Stretches [27 min.]
- Graphing Techniques: Reflections Across the Axes [17 min.]
- Putting It All Together: Moving The Graphs of Functions Around The Plane [36 min.]
Tape 13
- The Algebra of Functions [16 min.]
- A New Operation Unique To Functions: Composition [24 min.]
Tape 14
- Mathematical Models of Real World Problems: Constructing Functions [55 min.]
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UNIT 3 - EQUATIONS & INEQUALITIES
[7 hr. 16 min.]
Tape 15
EQUATIONS IN 1 VARIABLE [4 hr. 38 min.]
- Solving Equations (Approximately) with a Graphing Device & The Intermediate Value Theorem [44 min.]
- Solving Linear Equations: The "Linear Formula" or, Graphing [23 min.]
- Solving Non-Linear Equations That Lead To Linear Equations [13 min.]
Tape 16
- Solving Quadratic Equations: Factoring, or, Graphing [11 min.]
- A Complex Reminder & The Principal Square Root of a Negative Number [12 min.]
- Solving Quadratic Equations: The "Quadratic Formula" & The Discriminant or, Graphing [50 min.]
Tape 17
- Some Linear & Quadratic Equation Exercises Explained [24 min.]
Tape 18
- Setting Up Equations: More Mathematical Models [65 min.]
Tape 19
- Solving "Radical" Equations [14 min.]
- Solving Equations "Quadratic in Form" [15 min.]
- Solving Factorable Equations [7 min.]
Tape 20
INEQUALITIES IN 1 VARIABLE [1 hr. 54 min.]
- Properties of Inequalities [20 min.]
- Solving Inequalities In General [17 min.]
- Solving Linear Inequalities [19 min.]
Tape 21
- Solving Quadratic Inequalities [17 min.]
- Solving Higher-Degree Polynomial Inequalities [14 min.]
- Solving Rational Inequalities [27 min.]
Tape 22
EQUATIONS & INEQUALITIES IN 1 VARIABLE [44 min.]
- When Absolute Value Appears: Equations [12 min.]
- When Absolute Value Appears: Inequalities [17 min.]
- More Exercises Explained [15 min.]
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UNIT 4 - POLYNOMIAL & RATIONAL FUNCTIONS
[5 hr. 23 min.]
Tape 23
POLYNOMIAL FUNCTIONS [2 hr. 30 min.]
- Degree 2: Quadratic Functions [19 min.]
- Graphing Quadratic Functions [39 min.]
- Quadratic Functions As Mathematical Models [25 min.]
Tape 24
- Degree "n": General Polynomial Functions [8 min.]
- Special Case: Power Functions & Their Graphs [18 min.]
- Graphing General Polynomial Functions: Zeros, Multiplicity, Turning Points & End Behavior [40 min.]
Tape 25
LOCATING THE ZEROS OF A POLYNOMIAL FUNCTION [1 hr. 28 min.]
- How Many Zeros Are There? [12 min.]
- How Many Zeros Are Real? [13 min.]
- How Many Real Zeros Are Positive? Negative? [16 min.]
- Where (On What Interval) Are All Real Zeros? [22 min.]
- How Can You Guess The Location of Real Zeros? [4 min.]
- How Can You Reduce The Number of Real Zeros? [21 min.]
Tape 26
- Strategy & Tools: A Practical Checklist [14 min.]
- Some Polynomial Exercises Explained [22 min.]
Tape 27
RATIONAL FUNCTIONS [1 hr. 25 min.]
- General Rational Functions [14 min.]
- What Is An Asymptote? [17 min.]
- Finding Asymptotes of Rational Functions [35 min.]
- Graphing Rational Functions [19 min.]
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UNIT 5 - EXPONENTIAL & LOGARITHMIC FUNCTIONS
[3 hr. 42 min.]
Tape 28
EXPONENTIAL FUNCTIONS [55 min.]
- One-To-One Functions [14 min.]
- Exponential Functions & Their Graphs [26 min.]
- The Natural Exponential Function [15 min.]
Tape 29
LOGARITHMIC FUNCTIONS [1 hr. 59 min.]
- Inverse Functions [23 min.]
- Logarithmic Functions & Their Graphs [24 min.]
- The Natural Logarithmic Function [4 min.]
Tape 30
- Properties of Logarithms [17 min.]
- All Logarithms are Natural (or Common)! [15 min.]
- Solving Logarithmic Equations [20 min.]
- Logarithmic Models: Sound (Loudness) & Fury (Earthquakes) [16 min.]
Tape 31
EXPONENTIAL FUNCTIONS (con.) [47 min.]
- Solving Exponential Equations [15 min.]
- Exponential Models: Compounded Interest, and Growth & Decay [32 min.]
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UNIT 6 - SYSTEMS OF EQUATIONS
[2 hr. 44 min.]
Tape 32
SYSTEMS OF LINEAR EQUATIONS [2 hr. 29 min.]
- Systems of Linear Equations in General [19 min.]
- Solving A System of 2 (or 3) Linear Equations in 2 (or 3) Variables: Substitution [20 min.]
- Solving A System of 2 (or 3) Linear Equations in 2 (or 3) Variables: Elimination [36 min.]
Tape 33
- Some Exercises Explained [31 min.]
- An Application: Writing Proper Rational Functions As Sums of Simpler Proper Rational Functions (Partial Fractions) [43 min.]
Tape 34
SYSTEMS OF NON-LINEAR EQUATIONS [14 min]
- Solving (Mostly Graphically) a System of 2 Non-Linear Equations in 2 Variables [14 min.]
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UNIT 7 - SOME DISCRETE TOPICS
[3 hr. 40 min.]
Tape 35
SEQUENCES [1 hr. 41 min.]
- Infinite Sequences: Functions With Domain N [28 min.]
- The Factorial Symbol: ! [8 min.]
- Adding The First n Terms of a Sequence: nth Partial Sums & Summation Notation [17 min.]
Tape 36
- Arithmetic Sequences: Adding Your Way to Infinity [18 min.]
- Geometric Sequences: Multiplying Your Way to Infinity [30 min.]
Tape 37
SERIES & INDUCTION [56 min.]
- Geometric Series & Their (Infinite) Sums [26 min.]
- The Principle of Mathematical Induction [30 min.]
Tape 38
THE BINOMIAL THEOREM [1 hr. 2 min.]
- The "Binomial Coefficient" Symbol [11 min.]
- Pascal's Triangle [19 min.]
- The Binomial Theorem: How to Expand (x+a)n [32 min.]
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THE X-Y FILES: The Proof is in Here
[1 hr. 27 min.]
Tape 39
FOR UNIT 0 [29 min.]
- A Proof That Ö2 Is Irrational [6 min.]
- A Proof That There Are The Same Number of Rational Numbers as Natural Numbers! [9 min.]
- A Proof That There Are More Real Numbers Than Natural Numbers! [14 min.]
FOR UNIT 2 [17 min.]
- Algebra For Science: Variation [17 min.]
FOR UNIT 4 [21 min.]
- How Can You Find Rational Zeros (if any) of a Polynomial Function? [21 min.]
FOR UNIT 7 [19 min.]
- A Proof That "e" Is Irrational [19 min.]
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