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INSTRUCTORS Carleen Eaton Grant Fraser
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Dr. Carleen Eaton

Dr. Carleen Eaton

Solving Polynomial Functions

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Table of Contents

I. Equations and Inequalities
Expressions and Formulas

22m 23s

Intro
0:00
Order of Operations
0:19
Variable
0:27
Algebraic Expression
0:46
Term
0:57
Example: Algebraic Expression
1:25
Evaluate Inside Grouping Symbols
1:55
Evaluate Powers
2:30
Multiply/Divide Left to Right
2:55
Add/Subtract Left to Right
3:35
Monomials
4:40
Examples of Monomials
4:52
Constant
5:27
Coefficient
5:46
Degree
6:25
Power
7:15
Polynomials
8:02
Examples of Polynomials
8:24
Binomials, Trinomials, Monomials
8:53
Term
9:21
Like Terms
10:02
Formulas
11:00
Example: Pythagorean Theorem
11:15
Example 1: Evaluate the Algebraic Expression
11:50
Example 2: Evaluate the Algebraic Expression
14:38
Example 3: Area of a Triangle
19:11
Example 4: Fahrenheit to Celsius
20:41
Properties of Real Numbers

20m 15s

Intro
0:00
Real Numbers
0:07
Number Line
0:15
Rational Numbers
0:46
Irrational Numbers
2:24
Venn Diagram of Real Numbers
4:03
Irrational Numbers
5:00
Rational Numbers
5:19
Real Number System
5:27
Natural Numbers
5:32
Whole Numbers
5:53
Integers
6:19
Fractions
6:46
Properties of Real Numbers
7:15
Commutative Property
7:34
Associative Property
8:07
Identity Property
9:04
Inverse Property
9:53
Distributive Property
11:03
Example 1: What Set of Numbers?
12:21
Example 2: What Properties Are Used?
13:56
Example 3: Multiplicative Inverse
16:00
Example 4: Simplify Using Properties
17:18
Solving Equations

19m 10s

Intro
0:00
Translations
0:06
Verbal Expressions and Algebraic Expressions
0:13
Example: Sum of Two Numbers
0:19
Example: Square of a Number
1:33
Properties of Equality
3:20
Reflexive Property
3:30
Symmetric Property
3:42
Transitive Property
4:01
Addition Property
5:01
Subtraction Property
5:37
Multiplication Property
6:02
Division Property
6:30
Solving Equations
6:58
Example: Using Properties
7:18
Solving for a Variable
8:25
Example: Solve for Z
8:34
Example 1: Write Algebraic Expression
10:15
Example 2: Write Verbal Expression
11:31
Example 3: Solve the Equation
14:05
Example 4: Simplify Using Properties
17:26
Solving Absolute Value Equations

17m 31s

Intro
0:00
Absolute Value Expressions
0:09
Distance from Zero
0:18
Example: Absolute Value Expression
0:24
Absolute Value Equations
1:50
Example: Absolute Value Equation
2:00
Example: Isolate Expression
3:13
No Solution
3:46
Empty Set
3:58
Example: No Solution
4:12
Number of Solutions
4:46
Check Each Solution
4:57
Example: Two Solutions
5:05
Example: No Solution
6:18
Example: One Solution
6:28
Example 1: Evaluate for X
7:16
Example 2: Write Verbal Expression
9:08
Example 3: Solve the Equation
12:18
Example 4: Simplify Using Properties
13:36
Solving Inequalities

17m 14s

Intro
0:00
Properties of Inequalities
0:08
Addition Property
0:17
Example: Using Numbers
0:30
Subtraction Property
1:03
Example: Using Numbers
1:19
Multiplication Properties
1:44
C>0 (Positive Number)
1:50
Example: Using Numbers
2:05
C<0 (Negative Number)
2:40
Example: Using Numbers
3:10
Division Properties
4:11
C>0 (Positive Number)
4:15
Example: Using Numbers
4:27
C<0 (Negative Number)
5:21
Example: Using Numbers
5:32
Describing the Solution Set
6:10
Example: Set Builder Notation
6:26
Example: Graph (Closed Circle)
7:08
Example: Graph (Open Circle)
7:30
Example 1: Solve the Inequality
7:58
Example 2: Solve the Inequality
9:06
Example 3: Solve the Inequality
10:10
Example 4: Solve the Inequality
13:12
Solving Compound and Absolute Value Inequalities

25m

Intro
0:00
Compound Inequalities
0:08
And and Or
0:13
Example: And
0:22
Example: Or
1:12
And Inequality
1:41
Intersection
1:49
Example: Numbers
2:08
Example: Inequality
2:43
Or Inequality
4:35
Example: Union
4:45
Example: Inequality
5:53
Absolute Value Inequalities
7:19
Definition of Absolute Value
7:33
Examples: Compound Inequalities
8:30
Example: Complex Inequality
12:21
Example 1: Solve the Inequality
12:54
Example 2: Solve the Inequality
17:21
Example 3: Solve the Inequality
18:54
Example 4: Solve the Inequality
22:15
II. Linear Relations and Functions
Relations and Functions

32m 5s

Intro
0:00
Coordinate Plane
0:20
X-Coordinate and Y-Coordinate
0:30
Example: Coordinate Pairs
0:37
Quadrants
1:20
Relations
2:14
Domain and Range
2:19
Set of Ordered Pairs
2:29
As a Table
2:51
Functions
4:21
One Element in Range
4:32
Example: Mapping
4:43
Example: Table and Map
6:26
One-to-One Functions
8:01
Example: One-to-One
8:22
Example: Not One-to-One
9:18
Graphs of Relations
11:01
Discrete and Continuous
11:12
Example: Discrete
11:22
Example: Continous
12:30
Vertical Line Test
14:09
Example: S Curve
14:29
Example: Function
16:15
Equations, Relations, and Functions
17:03
Independent Variable and Dependent Variable
17:16
Function Notation
19:11
Example: Function Notation
19:23
Example 1: Domain and Range
20:51
Example 2: Discrete or Continous
23:03
Example 3: Discrete or Continous
25:53
Example 4: Function Notation
30:05
Linear Equations

14m 46s

Intro
0:00
Linear Equations and Functions
0:07
Linear Equation
0:19
Example: Linear Equation
0:29
Example: Linear Function
1:07
Standard Form
2:02
Integer Constants with No Common Factor
2:08
Example: Standard Form
2:27
Graphing with Intercepts
4:05
X-Intercept and Y-Intercept
4:12
Example: Intercepts
4:26
Example: Graphing
5:14
Example 1: Linear Function
7:53
Example 2: Linear Function
9:10
Example 3: Standard Form
10:04
Example 4: Graph with Intercepts
12:25
Slope

23m 7s

Intro
0:00
Definition of Slope
0:07
Change in Y / Change in X
0:26
Example: Slope of Graph
0:37
Interpretation of Slope
3:07
Horizontal Line (0 Slope)
3:13
Vertical Line (Undefined Slope)
4:52
Rises to Right (Positive Slope)
6:36
Falls to Right (Negative Slope)
6:53
Parallel Lines
7:18
Example: Not Vertical
7:30
Example: Vertical
7:58
Perpendicular Lines
8:31
Example: Perpendicular
8:42
Example 1: Slope of Line
10:32
Example 2: Graph Line
11:45
Example 3: Parallel to Graph
13:37
Example 4: Perpendicular to Graph
17:57
Writing Linear Functions

23m 5s

Intro
0:00
Slope Intercept Form
0:11
m and b
0:28
Example: Graph Using Slope Intercept
0:43
Point Slope Form
2:41
Relation to Slope Formula
3:03
Example: Point Slope Form
4:36
Parallel and Perpendicular Lines
6:28
Review of Parallel and Perpendicular Lines
6:31
Example: Parallel
7:50
Example: Perpendicular
9:58
Example 1: Slope Intercept Form
11:07
Example 2: Slope Intercept Form
13:07
Example 3: Parallel
15:49
Example 4: Perpendicular
18:42
Special Functions

31m 5s

Intro
0:00
Step Functions
0:07
Example: Apple Prices
0:30
Absolute Value Function
4:55
Example: Absolute Value
5:05
Piecewise Functions
9:08
Example: Piecewise
9:27
Example 1: Absolute Value Function
14:00
Example 2: Absolute Value Function
20:39
Example 3: Piecewise Function
22:26
Example 4: Step Function
25:25
Graphing Inequalities

21m 42s

Intro
0:00
Graphing Linear Inequalities
0:07
Shaded Region
0:19
Using Test Points
0:32
Graph Corresponding Linear Function
0:46
Dashed or Solid Lines
0:59
Use Test Point
1:21
Example: Linear Inequality
1:58
Graphing Absolute Value Inequalities
4:50
Graph Corresponding Equations
4:59
Use Test Point
5:20
Example: Absolute Value Inequality
5:38
Example 1: Linear Inequality
9:17
Example 2: Linear Inequality
11:56
Example 3: Linear Inequality
14:29
Example 4: Absolute Value Inequality
17:06
III. Systems of Equations and Inequalities
Solving Systems of Equations by Graphing

17m 13s

Intro
0:00
Systems of Equations
0:09
Example: Two Equations
0:24
Solving by Graphing
0:53
Point of Intersection
1:09
Types of Systems
2:29
Independent (Single Solution)
2:34
Dependent (Infinite Solutions)
3:05
Inconsistent (No Solution)
4:23
Example 1: Solve by Graphing
5:20
Example 2: Solve by Graphing
9:10
Example 3: Solve by Graphing
12:27
Example 4: Solve by Graphing
14:54
Solving Systems of Equations Algebraically

23m 53s

Intro
0:00
Solving by Substitution
0:08
Example: System of Equations
0:36
Solving by Multiplication
7:22
Extra Step of Multiplying
7:38
Example: System of Equations
8:00
Inconsistent and Dependent Systems
11:14
Variables Drop Out
11:48
Inconsistent System (Never True)
12:01
Constant Equals Constant
12:53
Dependent System (Always True)
13:11
Example 1: Solve Algebraically
13:58
Example 2: Solve Algebraically
15:52
Example 3: Solve Algebraically
17:54
Example 4: Solve Algebraically
21:40
Solving Systems of Inequalities By Graphing

27m 12s

Intro
0:00
Solving by Graphing
0:08
Graph Each Inequality
0:25
Overlap
0:35
Corresponding Linear Equations
1:03
Test Point
1:23
Example: System of Inequalities
1:51
No Solution
7:06
Empty Set
7:26
Example: No Solution
7:34
Example 1: Solve by Graphing
10:27
Example 2: Solve by Graphing
13:30
Example 3: Solve by Graphing
17:19
Example 4: Solve by Graphing
23:23
Solving Systems of Equations in Three Variables

28m 53s

Intro
0:00
Solving Systems in Three Variables
0:17
Triple of Values
0:31
Example: Three Variables
0:56
Number of Solutions
5:55
One Solution
6:08
No Solution
6:24
Infinite Solutions
7:06
Example 1: Solve 3 Variables
7:59
Example 2: Solve 3 Variables
13:50
Example 3: Solve 3 Variables
19:54
Example 4: Solve 3 Variables
25:50
IV. Matrices
Basic Matrix Concepts

11m 34s

Intro
0:00
What is a Matrix
0:26
Brackets
0:46
Designation
1:21
Element
1:47
Matrix Equations
1:59
Dimensions
2:27
Rows (m) and Columns (n)
2:37
Examples: Dimensions
2:43
Special Matrices
4:22
Row Matrix
4:32
Column Matrix
5:00
Zero Matrix
6:00
Equal Matrices
6:30
Example: Corresponding Elements
6:36
Example 1: Matrix Dimension
8:12
Example 2: Matrix Dimension
9:03
Example 3: Zero Matrix
9:38
Example 4: Row and Column Matrix
10:26
Matrix Operations

21m 36s

Intro
0:00
Matrix Addition
0:18
Same Dimensions
0:25
Example: Adding Matrices
1:04
Matrix Subtraction
3:42
Same Dimensions
3:48
Example: Subtracting Matrices
4:04
Scalar Multiplication
6:08
Scalar Constant
6:24
Example: Multiplying Matrices
6:32
Properties of Matrix Operations
8:23
Commutative Property
8:41
Associative Property
9:08
Distributive Property
9:44
Example 1: Matrix Addition
10:24
Example 2: Matrix Subtraction
11:58
Example 3: Scalar Multiplication
14:23
Example 4: Matrix Properties
16:09
Matrix Multiplication

29m 36s

Intro
0:00
Dimension Requirement
0:17
n = p
0:24
Resulting Product Matrix (m x q)
1:21
Example: Multiplication
1:54
Matrix Multiplication
3:38
Example: Matrix Multiplication
4:07
Properties of Matrix Multiplication
10:46
Associative Property
11:00
Associative Property (Scalar)
11:28
Distributive Property
12:06
Distributive Property (Scalar)
12:30
Example 1: Possible Matrices
13:31
Example 2: Multiplying Matrices
17:08
Example 3: Multiplying Matrices
20:41
Example 4: Matrix Properties
24:41
Determinants

33m 13s

Intro
0:00
What is a Determinant
0:13
Square Matrices
0:23
Vertical Bars
0:41
Determinant of a 2x2 Matrix
1:21
Second Order Determinant
1:37
Formula
1:45
Example: 2x2 Determinant
1:58
Determinant of a 3x3 Matrix
2:50
Expansion by Minors
3:08
Third Order Determinant
3:19
Expanding Row One
4:06
Example: 3x3 Determinant
6:40
Diagonal Method for 3x3 Matrices
13:24
Example: Diagonal Method
13:36
Example 1: Determinant of 2x2
18:59
Example 2: Determinant of 3x3
20:03
Example 3: Determinant of 3x3
25:35
Example 4: Determinant of 3x3
29:22
Cramer's Rule

28m 25s

Intro
0:00
System of Two Equations in Two Variables
0:16
One Variable
0:50
Determinant of Denominator
1:14
Determinants of Numerators
2:23
Example: System of Equations
3:34
System of Three Equations in Three Variables
7:06
Determinant of Denominator
7:17
Determinants of Numerators
7:52
Example 1: Two Equations
8:57
Example 2: Two Equations
13:21
Example 3: Three Equations
17:11
Example 4: Three Equations
23:43
Identity and Inverse Matrices

22m 25s

Intro
0:00
Identity Matrix
0:13
Example: 2x2 Identity Matrix
0:30
Example: 4x4 Identity Matrix
0:50
Properties of Identity Matrices
1:24
Example: Multiplying Identity Matrix
2:52
Matrix Inverses
5:30
Writing Matrix Inverse
6:07
Inverse of a 2x2 Matrix
6:39
Example: 2x2 Matrix
7:31
Example 1: Inverse Matrix
10:18
Example 2: Find the Inverse Matrix
13:04
Example 3: Find the Inverse Matrix
17:53
Example 4: Find the Inverse Matrix
20:44
Solving Systems of Equations Using Matrices

22m 32s

Intro
0:00
Matrix Equations
0:11
Example: System of Equations
0:21
Solving Systems of Equations
4:01
Isolate x
4:16
Example: Using Numbers
5:10
Multiplicative Inverse
5:54
Example 1: Write as Matrix Equation
7:18
Example 2: Use Matrix Equations
9:12
Example 3: Use Matrix Equations
15:06
Example 4: Use Matrix Equations
19:35
V. Quadratic Functions and Inequalities
Graphing Quadratic Functions

31m 48s

Intro
0:00
Quadratic Functions
0:12
A is Zero
0:27
Example: Parabola
0:45
Properties of Parabolas
2:08
Axis of Symmetry
2:11
Vertex
2:32
Example: Parabola
2:48
Minimum and Maximum Values
9:02
Positive or Negative
9:28
Upward or Downward
9:58
Example: Minimum
10:31
Example: Maximum
11:16
Example 1: Axis of Symmetry, Vertex, Graph
12:41
Example 2: Axis of Symmetry, Vertex, Graph
17:25
Example 3: Minimum or Maximum
21:47
Example 4: Minimum or Maximum
27:09
Solving Quadratic Equations by Graphing

27m 3s

Intro
0:00
Quadratic Equations
0:16
Standard Form
0:18
Example: Quadratic Equation
0:47
Solving by Graphing
1:41
Roots (x-Intercepts)
1:48
Example: Number of Solutions
2:12
Estimating Solutions
9:23
Example: Integer Solutions
9:30
Example: Estimating
9:53
Example 1: Solve by Graphing
10:52
Example 2: Solve by Graphing
15:10
Example 1: Solve by Graphing
17:50
Example 1: Solve by Graphing
20:54
Solving Quadratic Equations by Factoring

19m 53s

Intro
0:00
Factoring Techniques
0:15
Greatest Common Factor (GCF)
0:37
Difference of Two Squares
1:48
Perfect Square Trinomials
2:30
General Trinomials
3:09
Zero Product Rule
5:22
Example: Zero Product
5:53
Example 1: Solve by Factoring
7:46
Example 1: Solve by Factoring
9:48
Example 1: Solve by Factoring
12:34
Example 1: Solve by Factoring
15:28
Imaginary and Complex Numbers

35m 45s

Intro
0:00
Properties of Square Roots
0:10
Product Property
0:26
Example: Product Property
0:56
Quotient Property
2:17
Example: Quotient Property
2:35
Imaginary Numbers
3:12
Imaginary i
3:51
Examples: Imaginary Number
4:22
Complex Numbers
7:23
Real Part and Imaginary Part
7:33
Examples: Complex Numbers
7:57
Equality
9:37
Example: Equal Complex Numbers
9:52
Addition and Subtraction
10:12
Examples: Adding Complex Numbers
10:25
Complex Plane
13:32
Horizontal Axis (Real)
13:49
Vertical Axis (Imaginary)
13:59
Example: Labeling
14:11
Multiplication
15:57
Example: FOIL Method
16:03
Division
18:37
Complex Conjugates
18:45
Conjugate Pairs
19:10
Example: Dividing Complex Numbers
20:00
Example 1: Simplify Complex Number
24:50
Example 2: Simplify Complex Number
27:56
Example 3: Multiply Complex Numbers
29:27
Example 3: Dividing Complex Numbers
31:48
Completing the Square

27m 11s

Intro
0:00
Square Root Property
0:12
Example: Perfect Square
0:38
Example: Perfect Square Trinomial
3:00
Completing the Square
4:39
Constant Term
4:50
Example: Complete the Square
5:04
Solve Equations
6:42
Add to Both Sides
6:59
Example: Complete the Square
7:07
Equations Where a Not Equal to 1
10:58
Divide by Coefficient
11:08
Example: Complete the Square
11:24
Complex Solutions
14:05
Real and Imaginary
14:14
Example: Complex Solution
14:35
Example 1: Square Root Property
18:31
Example 2: Complete the Square
19:15
Example 3: Complete the Square
20:40
Example 4: Complete the Square
23:56
Quadratic Formula and the Discriminant

22m 48s

Intro
0:00
Quadratic Formula
0:21
Standard Form
0:29
Example: Quadratic Formula
0:57
One Rational Root
3:00
Example: One Root
3:31
Complex Solutions
6:16
Complex Conjugate
6:28
Example: Complex Solution
7:15
Discriminant
9:42
Positive Discriminant
10:03
Perfect Square (Rational)
10:51
Not Perfect Square (2 Irrational)
11:27
Negative Discriminant
12:28
Zero Discriminant
12:57
Example 1: Quadratic Formula
13:50
Example 2: Quadratic Formula
16:03
Example 3: Quadratic Formula
19:00
Example 4: Discriminant
21:33
Analyzing the Graphs of Quadratic Functions

30m 7s

Intro
0:00
Vertex Form
0:12
H and K
0:32
Axis of Symmetry
0:36
Vertex
0:42
Example: Origin
1:00
Example: k = 2
2:12
Example: h = 1
4:27
Significance of Coefficient a
7:13
Example: |a| > 1
7:25
Example: |a| < 1
8:18
Example: |a| > 0
8:51
Example: |a| < 0
9:05
Writing Quadratic Equations in Vertex Form
10:22
Standard Form to Vertex Form
10:35
Example: Standard Form
11:02
Example: a Term Not 1
14:42
Example 1: Vertex Form
19:47
Example 2: Vertex Form
22:09
Example 3: Vertex Form
24:32
Example 4: Vertex Form
28:23
Graphing and Solving Quadratic Inequalities

27m 5s

Intro
0:00
Graphing Quadratic Inequalities
0:11
Test Point
0:18
Example: Quadratic Inequality
0:29
Solving Quadratic Inequalities
3:57
Example: Parameter
4:24
Example 1: Graph Inequality
11:16
Example 2: Solve Inequality
14:27
Example 3: Graph Inequality
19:14
Example 4: Solve Inequality
23:48
VI. Polynomial Functions
Properties of Exponents

19m 29s

Intro
0:00
Simplifying Exponential Expressions
0:09
Monomial Simplest Form
0:19
Negative Exponents
1:07
Examples: Simple
1:34
Properties of Exponents
3:06
Negative Exponents
3:13
Mutliplying Same Base
3:24
Dividing Same Base
3:45
Raising Power to a Power
4:33
Parentheses (Multiplying)
5:11
Parentheses (Dividing)
5:47
Raising to 0th Power
6:15
Example 1: Simplify Exponents
7:59
Example 2: Simplify Exponents
10:41
Example 3: Simplify Exponents
14:11
Example 4: Simplify Exponents
18:04
Operations on Polynomials

13m 27s

Intro
0:00
Adding and Subtracting Polynomials
0:13
Like Terms and Like Monomials
0:23
Examples: Adding Monomials
1:14
Multiplying Polynomials
3:40
Distributive Property
3:44
Example: Monomial by Polynomial
4:06
Example 1: Simplify Polynomials
5:47
Example 2: Simplify Polynomials
6:28
Example 3: Simplify Polynomials
8:38
Example 4: Simplify Polynomials
10:47
Dividing Polynomials

31m 11s

Intro
0:00
Dividing by a Monomial
0:13
Example: Numbers
0:26
Example: Polynomial by a Monomial
1:18
Long Division
2:28
Remainder Term
2:41
Example: Dividing with Numbers
3:04
Example: With Polynomials
5:01
Example: Missing Terms
7:58
Synthetic Division
11:44
Restriction
12:04
Example: Divisor in Form
12:20
Divisor in Synthetic Division
15:54
Example: Coefficient to 1
16:07
Example 1: Divide Polynomials
17:10
Example 2: Divide Polynomials
19:08
Example 3: Synthetic Division
21:42
Example 4: Synthetic Division
25:09
Polynomial Functions

22m 30s

Intro
0:00
Polynomial in One Variable
0:13
Leading Coefficient
0:27
Example: Polynomial
1:18
Degree
1:31
Polynomial Functions
2:57
Example: Function
3:13
Function Values
3:33
Example: Numerical Values
3:53
Example: Algebraic Expressions
5:11
Zeros of Polynomial Functions
5:50
Odd Degree
6:04
Even Degree
7:29
End Behavior
8:28
Even Degrees
9:09
Example: Leading Coefficient +/-
9:23
Odd Degrees
12:51
Example: Leading Coefficient +/-
13:00
Example 1: Degree and Leading Coefficient
15:03
Example 2: Polynomial Function
15:56
Example 3: Polynomial Function
17:34
Example 4: End Behavior
19:53
Analyzing Graphs of Polynomial Functions

33m 29s

Intro
0:00
Graphing Polynomial Functions
0:11
Example: Table and End Behavior
0:39
Location Principle
4:43
Zero Between Two Points
5:03
Example: Location Principle
5:21
Maximum and Minimum Points
8:40
Relative Maximum and Relative Minimum
9:16
Example: Number of Relative Max/Min
11:11
Example 1: Graph Polynomial Function
11:57
Example 2: Graph Polynomial Function
16:19
Example 3: Graph Polynomial Function
23:27
Example 4: Graph Polynomial Function
28:35
Solving Polynomial Functions

21m 10s

Intro
0:00
Factoring Polynomials
0:06
Greatest Common Factor (GCF)
0:25
Difference of Two Squares
1:14
Perfect Square Trinomials
2:07
General Trinomials
2:57
Grouping
4:32
Sum and Difference of Two Cubes
6:03
Examples: Two Cubes
6:14
Quadratic Form
8:22
Example: Quadratic Form
8:44
Example 1: Factor Polynomial
12:03
Example 2: Factor Polynomial
13:54
Example 3: Quadratic Form
15:33
Example 4: Solve Polynomial Function
17:24
Remainder and Factor Theorems

31m 21s

Intro
0:00
Remainder Theorem
0:07
Checking Work
0:22
Dividend and Divisor in Theorem
1:12
Example: f(a)
2:05
Synthetic Substitution
5:43
Example: Polynomial Function
6:15
Factor Theorem
9:54
Example: Numbers
10:16
Example: Confirm Factor
11:27
Factoring Polynomials
14:48
Example: 3rd Degree Polynomial
15:07
Example 1: Remainder Theorem
19:17
Example 2: Other Factors
21:57
Example 3: Remainder Theorem
25:52
Example 4: Other Factors
28:21
Roots and Zeros

31m 27s

Intro
0:00
Number of Roots
0:08
Not Nature of Roots
0:18
Example: Real and Complex Roots
0:25
Descartes' Rule of Signs
2:05
Positive Real Roots
2:21
Example: Positve
2:39
Negative Real Roots
5:44
Example: Negative
6:06
Finding the Roots
9:59
Example: Combination of Real and Complex
10:07
Conjugate Roots
13:18
Example: Conjugate Roots
13:50
Example 1: Solve Polynomial
16:03
Example 2: Solve Polynomial
18:36
Example 3: Possible Combinations
23:13
Example 4: Possible Combinations
27:11
Rational Zero Theorem

31m 16s

Intro
0:00
Equation
0:08
List of Possibilities
0:16
Equation with Constant and Leading Coefficient
1:04
Example: Rational Zero
2:46
Leading Coefficient Equal to One
7:19
Equation with Leading Coefficient of One
7:34
Example: Coefficient Equal to 1
8:45
Finding Rational Zeros
12:58
Division with Remainder Zero
13:32
Example 1: Possible Rational Zeros
14:20
Example 2: Possible Rational Zeros
16:02
Example 3: Possible Rational Zeros
19:58
Example 4: Find All Zeros
22:06
VII. Radical Expressions and Inequalities
Operations on Functions

34m 30s

Intro
0:00
Arithmetic Operations
0:07
Domain
0:16
Intersection
0:24
Denominator is Zero
0:49
Example: Operations
1:02
Composition of Functions
7:18
Notation
7:48
Right to Left
8:18
Example: Composition
8:48
Composition is Not Commutative
17:23
Example: Not Commutative
17:51
Example 1: Function Operations
20:55
Example 2: Function Operations
24:34
Example 3: Compositions
27:51
Example 4: Function Operations
31:09
Inverse Functions and Relations

22m 42s

Intro
0:00
Inverse of a Relation
0:14
Example: Ordered Pairs
0:56
Inverse of a Function
3:24
Domain and Range Switched
3:52
Example: Inverse
4:28
Procedure to Construct an Inverse Function
6:42
f(x) to y
6:42
Interchange x and y
6:59
Solve for y
7:06
Write Inverse f(x) for y
7:14
Example: Inverse Function
7:25
Example: Inverse Function 2
8:48
Inverses and Compositions
10:44
Example: Inverse Composition
11:46
Example 1: Inverse Relation
14:49
Example 2: Inverse of Function
15:40
Example 3: Inverse of Function
17:06
Example 4: Inverse Functions
18:55
Square Root Functions and Inequalities

30m 4s

Intro
0:00
Square Root Functions
0:07
Examples: Square Root Function
0:16
Example: Not Square Root Function
0:46
Radicand
1:12
Example: Restriction
1:31
Graphing Square Root Functions
3:42
Example: Graphing
3:49
Square Root Inequalities
8:47
Same Technique
9:00
Example: Square Root Inequality
9:20
Example 1: Graph Square Root Function
15:19
Example 2: Graph Square Root Function
18:03
Example 3: Graph Square Root Function
22:41
Example 4: Square Root Inequalities
25:37
nth Roots

20m 46s

Intro
0:00
Definition of the nth Root
0:07
Example: 5th Root
0:20
Example: 6th Root
0:51
Principal nth Root
1:39
Example: Principal Roots
2:06
Using Absolute Values
5:58
Example: Square Root
6:18
Example: 6th Root
8:40
Example: Negative
10:15
Example 1: Simplify Radicals
12:23
Example 2: Simplify Radicals
13:29
Example 3: Simplify Radicals
16:07
Example 4: Simplify Radicals
18:18
Operations with Radical Expressions

41m 11s

Intro
0:00
Properties of Radicals
0:16
Quotient Property
0:29
Example: Quotient
1:00
Example: Product Property
1:47
Simplifying Radical Expressions
3:24
Radicand No nth Powers
3:47
Radicand No Fractions
6:33
No Radicals in Denominator
7:16
Rationalizing Denominators
8:27
Example: Radicand nth Power
9:05
Conjugate Radical Expressions
11:47
Conjugates
12:07
Example: Conjugate Radical Expression
13:11
Adding and Subtracting Radicals
16:12
Same Index, Same Radicand
16:20
Example: Like Radicals
16:28
Multiplying Radicals
19:04
Distributive Property
19:10
Example: Multiplying Radicals
19:20
Example 1: Simplify Radical
24:11
Example 2: Simplify Radicals
28:43
Example 3: Simplify Radicals
32:00
Example 4: Simplify Radical
36:34
Rational Exponents

30m 45s

Intro
0:00
Definition 1
0:20
Example: Using Numbers
0:39
Example: Non-Negative
2:46
Example: Odd
3:34
Definition 2
4:32
Restriction
4:52
Example: Relate to Definition 1
5:04
Example: m Not 1
5:31
Simplifying Expressions
7:53
Multiplication
8:31
Division
9:29
Multiply Exponents
10:08
Raised Power
11:05
Zero Power
11:29
Negative Power
11:49
Simplified Form
13:52
Complex Fraction
14:16
Negative Exponents
14:40
Example: More Complicated
15:14
Example 1: Write as Radical
19:03
Example 2: Write with Rational Exponents
20:40
Example 3: Complex Fraction
22:09
Example 4: Complex Fraction
26:22
Solving Radical Equations and Inequalities

31m 27s

Intro
0:00
Radical Equations
0:11
Variables in Radicands
0:22
Example: Radical Equation
1:06
Example: Complex Equation
2:42
Extraneous Roots
7:21
Squaring Technique
7:35
Double Check
7:44
Example: Extraneous
8:21
Eliminating nth Roots
10:04
Isolate and Raise Power
10:14
Example: nth Root
10:27
Radical Inequalities
11:27
Restriction: Index is Even
11:53
Example: Radical Inequality
12:29
Example 1: Solve Radical Equation
15:41
Example 2: Solve Radical Equation
17:44
Example 3: Solve Radical Inequality
20:24
Example 4: Solve Radical Equation
24:34
VIII. Rational Equations and Inequalities
Multiplying and Dividing Rational Expressions

40m 54s

Intro
0:00
Simplifying Rational Expressions
0:22
Algebraic Fraction
0:29
Examples: Rational Expressions
0:49
Example: GCF
1:33
Example: Simplify Rational Expression
2:26
Factoring -1
4:04
Example: Simplify with -1
4:19
Multiplying and Dividing Rational Expressions
6:59
Multiplying and Dividing
7:28
Example: Multiplying Rational Expressions
8:36
Example: Dividing Rational Expressions
11:20
Factoring
14:01
Factoring Polynomials
14:19
Example: Factoring
14:35
Complex Fractions
18:22
Example: Numbers
18:37
Example: Algebraic Complex Fractions
19:25
Example 1: Simplify Rational Expression
25:56
Example 2: Simplify Rational Expression
29:34
Example 3: Simplify Rational Expression
31:39
Example 4: Simplify Rational Expression
37:50
Adding and Subtracting Rational Expressions

55m 4s

Intro
0:00
Least Common Multiple (LCM)
0:27
Examples: LCM of Numbers
0:43
Example: LCM of Polynomials
4:02
Adding and Subtracting
7:55
Least Common Denominator (LCD)
8:07
Example: Numbers
8:17
Example: Rational Expressions
11:03
Equivalent Fractions
15:22
Simplifying Complex Fractions
21:19
Example: Previous Lessons
21:36
Example: More Complex
22:53
Example 1: Find LCM
28:30
Example 2: Add Rational Expressions
31:44
Example 3: Subtract Rational Expressions
39:18
Example 4: Simplify Rational Expression
38:26
Graphing Rational Functions

57m 13s

Intro
0:00
Rational Functions
0:18
Restriction
0:34
Example: Rational Function
0:51
Breaks in Continuity
2:52
Example: Continuous Function
3:10
Discontinuities
3:30
Example: Excluded Values
4:37
Graphs and Discontinuities
5:02
Common Binomial Factor (Hole)
5:08
Example: Common Factor
5:31
Asymptote
10:06
Example: Vertical Asymptote
11:08
Horizontal Asymptotes
20:00
Example: Horizontal Asymptote
20:25
Example 1: Holes and Vertical Asymptotes
26:12
Example 2: Graph Rational Faction
28:35
Example 3: Graph Rational Faction
39:23
Example 4: Graph Rational Faction
47:28
Direct, Joint, and Inverse Variation

20m 21s

Intro
0:00
Direct Variation
0:07
Constant of Variation
0:25
Graph of Constant Variation
1:26
Slope is Constant k
1:35
Example: Straight Lines
1:41
Joint Variation
2:48
Three Variables
2:52
Inverse Variation
3:38
Rewritten Form
3:52
Examples in Biology
4:22
Graph of Inverse Variation
4:51
Asymptotes are Axes
5:12
Example: Inverse Variation
5:40
Proportions
10:11
Direct Variation
10:25
Inverse Variation
11:32
Example 1: Type of Variation
12:42
Example 2: Direct Variation
14:13
Example 3: Joint Variation
16:24
Example 4: Graph Rational Faction
18:50
Solving Rational Equations and Inequalities

55m 14s

Intro
0:00
Rational Equations
0:15
Example: Algebraic Fraction
0:26
Least Common Denominator
0:49
Example: Simple Rational Equation
1:22
Example: Solve Rational Equation
5:40
Extraneous Solutions
9:31
Doublecheck
10:00
No Solution
10:38
Example: Extraneous
10:44
Rational Inequalities
14:01
Excluded Values
14:31
Solve Related Equation
14:49
Find Intervals
14:58
Use Test Values
15:25
Example: Rational Inequality
15:51
Example: Rational Inequality 2
17:07
Example 1: Rational Equation
28:50
Example 2: Rational Equation
33:51
Example 3: Rational Equation
38:19
Example 4: Rational Inequality
46:49
IX. Exponential and Logarithmic Relations
Exponential Functions

35m 58s

Intro
0:00
What is an Exponential Function?
0:12
Restriction on b
0:31
Base
0:46
Example: Exponents as Bases
0:56
Variables as Exponents
1:12
Example: Exponential Function
1:50
Graphing Exponential Functions
2:33
Example: Using Table
2:49
Properties
11:52
Continuous and One to One
12:00
Domain is All Real Numbers
13:14
X-Axis Asymptote
13:55
Y-Intercept
14:02
Reflection Across Y-Axis
14:31
Growth and Decay
15:06
Exponential Growth
15:10
Real Life Examples
15:41
Example: Growth
15:52
Example: Decay
16:12
Real Life Examples
16:30
Equations
17:32
Bases are Same
18:05
Examples: Variables as Exponents
18:20
Inequalities
21:29
Property
21:51
Example: Inequality
22:37
Example 1: Graph Exponential Function
24:05
Example 2: Growth or Decay
27:50
Example 3: Exponential Equation
29:31
Example 4: Exponential Inequality
32:54
Logarithms and Logarithmic Functions

45m 54s

Intro
0:00
What are Logarithms?
0:08
Restrictions
0:15
Written Form
0:26
Logarithms are Exponents
0:52
Example: Logarithms
1:49
Logarithmic Functions
5:14
Same Restrictions
5:30
Inverses
5:53
Example: Logarithmic Function
6:24
Graph of the Logarithmic Function
9:20
Example: Using Table
9:35
Properties
15:09
Continuous and One to One
15:14
Domain
15:36
Range
15:56
Y-Axis is Asymptote
16:02
X Intercept
16:12
Inverse Property
16:57
Compositions of Functions
17:10
Equations
18:30
Example: Logarithmic Equation
19:13
Inequalities
20:36
Properties
20:47
Example: Logarithmic Inequality
21:40
Equations with Logarithms on Both Sides
24:43
Property
24:51
Example: Both Sides
25:23
Inequalities with Logarithms on Both Sides
26:52
Property
27:02
Example: Both Sides
28:05
Example 1: Solve Log Equation
31:52
Example 2: Solve Log Equation
33:53
Example 3: Solve Log Equation
36:15
Example 4: Solve Log Inequality
39:19
Properties of Logarithms

28m 43s

Intro
0:00
Product Property
0:08
Example: Product
0:46
Quotient Property
2:40
Example: Quotient
2:59
Power Property
3:51
Moved Exponent
4:07
Example: Power
4:37
Equations
5:15
Example: Use Properties
5:58
Example 1: Simplify Log
11:17
Example 2: Single Log
15:54
Example 3: Solve Log Equation
18:48
Example 4: Solve Log Equation
22:13
Common Logarithms

25m 23s

Intro
0:00
What are Common Logarithms?
0:10
Real World Applications
0:16
Base Not Written
0:27
Example: Base 10
0:39
Equations
1:47
Example: Same Base
1:56
Example: Different Base
2:37
Inequalities
6:07
Multiplying/Dividing Inequality
6:21
Example: Log Inequality
6:54
Change of Base
12:45
Base 10
13:24
Example: Change of Base
14:05
Example 1: Log Equation
15:21
Example 2: Common Logs
17:13
Example 3: Log Equation
18:22
Example 4: Log Inequality
21:52
Base e and Natural Logarithms

21m 14s

Intro
0:00
Number e
0:09
Natural Base
0:21
Growth/Decay
0:33
Example: Exponential Function
0:53
Natural Logarithms
1:11
ln x
1:19
Inverse and Identity Function
1:39
Example: Inverse Composition
1:55
Equations and Inequalities
4:39
Extraneous Solutions
5:30
Examples: Natural Log Equations
5:48
Example 1: Natural Log Equation
9:08
Example 2: Natural Log Equation
10:37
Example 3: Natural Log Inequality
16:54
Example 4: Natural Log Inequality
18:16
Exponential Growth and Decay

24m 30s

Intro
0:00
Decay
0:17
Decreases by Fixed Percentage
0:23
Rate of Decay
0:56
Example: Finance
1:34
Scientific Model of Decay
3:37
Exponential Decay
3:45
Radioactive Decay
4:13
Example: Half Life
5:33
Growth
9:06
Increases by Fixed Percentage
9:18
Example: Finance
10:09
Scientific Model of Growth
11:35
Population Growth
12:04
Example: Growth
12:20
Example 1: Computer Price
14:00
Example 2: Stock Price
15:46
Example 3: Medicine Disintegration
19:10
Example 4: Population Growth
22:33
X. Conic Sections
Midpoint and Distance Formulas

32m 42s

Intro
0:00
Midpoint Formula
0:15
Example: Midpoint
0:30
Distance Formula
2:30
Example: Distance
2:52
Example 1: Midpoint and Distance
4:58
Example 2: Midpoint and Distance
8:07
Example 3: Median Length
18:51
Example 4: Perimeter and Area
23:36
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
Radius
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
Example 2: Center and Radius
11:51
Example 3: Radius
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
Solving Quadratic Systems

47m 4s

Intro
0:00
Linear Quadratic Systems
0:22
Example: Linear Quadratic System
0:45
Solutions
2:49
Graphs of Possible Solutions
3:10
Quadratic Quadratic System
4:10
Example: Elimination
4:21
Solutions
11:39
Example: 0, 1, 2, 3, 4 Solutions
11:50
Systems of Quadratic Inequalities
12:48
Example: Quadratic Inequality
13:09
Example 1: Solve Quadratic System
21:42
Example 2: Solve Quadratic System
29:13
Example 3: Solve Quadratic System
35:02
Example 4: Solve Quadratic Inequality
40:29
XI. Sequences and Series
Arithmetic Sequences

21m 16s

Intro
0:00
Sequences
0:10
General Form of Sequence
0:16
Example: Finite/Infinite Sequences
0:33
Arithmetic Sequences
0:28
Common Difference
2:41
Example: Arithmetic Sequence
2:50
Formula for the nth Term
3:51
Example: nth Term
4:32
Equation for the nth Term
6:37
Example: Using Formula
6:56
Arithmetic Means
9:47
Example: Arithmetic Means
10:16
Example 1: nth Term
12:38
Example 2: Arithmetic Means
13:49
Example 3: Arithmetic Means
16:12
Example 4: nth Term
18:26
Arithmetic Series

21m 36s

Intro
0:00
What are Arithmetic Series?
0:11
Common Difference
0:28
Example: Arithmetic Sequence
0:43
Example: Arithmetic Series
1:09
Finite/Infinite Series
1:36
Sum of Arithmetic Series
2:27
Example: Sum
3:21
Sigma Notation
5:53
Index
6:14
Example: Sigma Notation
7:14
Example 1: First Term
9:00
Example 2: Three Terms
10:52
Example 3: Sum of Series
14:14
Example 4: Sum of Series
18:13
Geometric Sequences

23m 3s

Intro
0:00
Geometric Sequences
0:11
Common Difference
0:38
Common Ratio
1:08
Example: Geometric Sequence
2:38
nth Term of a Geometric Sequence
4:41
Example: nth Term
4:56
Geometric Means
6:51
Example: Geometric Mean
7:09
Example 1: 9th Term
12:04
Example 2: Geometric Means
15:18
Example 3: nth Term
18:32
Example 4: Three Terms
20:59
Geometric Series

22m 43s

Intro
0:00
What are Geometric Series?
0:11
List of Numbers
0:24
Example: Geometric Series
1:12
Sum of Geometric Series
2:16
Example: Sum of Geometric Series
2:41
Sigma Notation
4:21
Lower Index, Upper Index
4:38
Example: Sigma Notation
4:57
Another Sum Formula
6:08
Example: n Unknown
6:28
Specific Terms
7:41
Sum Formula
7:56
Example: Specific Term
8:11
Example 1: Sum of Geometric Series
10:02
Example 2: Sum of 8 Terms
14:15
Example 3: Sum of Geometric Series
18:23
Example 4: First Term
20:16
Infinite Geometric Series

18m 32s

Intro
0:00
What are Infinite Geometric Series
0:10
Example: Finite
0:29
Example: Infinite
0:51
Partial Sums
1:09
Formula
1:37
Sum of an Infinite Geometric Series
2:39
Convergent Series
2:58
Example: Sum of Convergent Series
3:28
Sigma Notation
7:31
Example: Sigma
8:17
Repeating Decimals
8:42
Example: Repeating Decimal
8:53
Example 1: Sum of Infinite Geometric Series
12:15
Example 2: Repeating Decimal
13:24
Example 3: Sum of Infinite Geometric Series
15:14
Example 4: Repeating Decimal
16:48
Recursion and Special Sequences

14m 34s

Intro
0:00
Fibonacci Sequence
0:05
Background of Fibonacci
0:23
Recursive Formula
0:37
Fibonacci Sequence
0:52
Example: Recursive Formula
2:18
Iteration
3:49
Example: Iteration
4:30
Example 1: Five Terms
7:08
Example 2: Three Terms
9:00
Example 3: Five Terms
10:38
Example 4: Three Iterates
12:41
Binomial Theorem

48m 30s

Intro
0:00
Pascal's Triangle
0:06
Expand Binomial
0:13
Pascal's Triangle
4:26
Properties
6:52
Example: Properties of Binomials
6:58
Factorials
9:11
Product
9:28
Example: Factorial
9:45
Binomial Theorem
11:08
Example: Binomial Theorem
13:48
Finding a Specific Term
18:36
Example: Specific Term
19:26
Example 1: Expand
24:39
Example 2: Fourth Term
30:26
Example 3: Five Terms
36:13
Example 4: Three Iterates
45:07
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Lecture Comments (5)

1 answer

Last reply by: Dr Carleen Eaton
Fri Jun 8, 2018 11:33 PM

Post by John Stedge on June 5 at 01:16:08 PM

In the last example you can factor the (x^2+4) into (x+2) (x+2)

1 answer

Last reply by: Dr Carleen Eaton
Tue Mar 13, 2018 7:15 PM

Post by MICHELLE AYOH on March 3 at 07:41:35 PM

very helpful ;)

0 answers

Post by Nsikan Esenowo on July 19, 2010

Thank you for all your effort put together on this site to make my Maths' problems solve. I learned a lot receiving lectures from you. You are the best that ever happen to me in this field. Maths made easy.

Solving Polynomial Functions

  • When factoring, always factor completely.
  • Always begin factoring by factoring the greatest common factor, and then factor the remaining expression.
  • If the polynomial has 4 terms, try factoring by grouping.
  • Factor trinomials by trial and error.
  • Memorize the formulas for the sum and difference of two cubes.

Solving Polynomial Functions

Factor 8x3 + y3
  • This problem is a Difference Of Cubes
  • 8x3 + y3 = (2x)3 + y3 because 2 to the third power = 8
  • The Pattern Looks like this:
  • x3 + y3 = (x + y)(x2 − xy + y2); where x = (2x) and y = y
  • Subsitute x and y into the pattern
  • (2x)3 + y3 = (2x + y)((2x)2 − (2x)y + y2)
  • Simplify Using Rules of Exponents
(2x)3 + y3 = (2x + y)(4x2 − 2xy + y2)
Factor 27a3 − 64b3
  • This problem is a Difference Of Cubes
  • 27a3 − 64b3 = (3a)3 − (4b)3 because 3 to the third power = 27 and 4 to the third power = 64
  • The Pattern Looks like this:
  • x3 − y3 = (x − y)(x2 + xy + y2); where x = (3a) and y = 4b
  • Subsitute x and y into the pattern
  • (3a)3 − (4b)3 = (3a − 4b)((3a)2 + (3a)(4b) + (4b)2)
  • Simplify Using Rules of Exponents
(3a)3 − (4b)3 = (3a − 4b)(9a2 + 123ab + 16b2)
Factor 6x3y3z3 − 8x4y3z4 − 2x2y3z3
  • Find the GCF, then cancel out anything you took out form each term.
  • Step 1: What is the GCF of 6, 8 and 2?
  • The GCF of those number is 2
  • Step 2:What is the maximum number of x's can you take out?
  • You can take out a maximum of 2 x, x2
  • Step 3: What is the maximum number of y's can you take out?
  • You can take out a maximum of 2 y′s, y2
  • Step 4:What is the maximum number of z's can you take out?
  • You can take out a maximum of 2 z′s, z2
  • What is the GCF then?
  • 2x2y2z2
  • Cancel out anything you took out form each term
2x2y2z2(3xyz − 4x2yz2 + yz)
Write in quadratic form: 3x4 − 2x2 − 21
  • Write the expression in the format ax2 + bx + c
  • 3(x2)2 − 2(x)2 − 21
  • Let p = x2
  • Notice that we can factor 3p2 − 2p − 21
  • ax2 + bx + c = 0
  • ax2 + bx + c = (x + [m/a])(x + [n/a])
  • Find two numbers "m" and "n" such that:
    a*c = m*n
    b = m + n
  • − 63 = m*n
  • − 2 = m + n
  • The only numbers that work is 7 and − 9
  • 3p2 − 2p − 21 = (p + [7/3])(p + [( − 9)/3]) = (3p + 7)(p − 3)
  • Given that p = x2, the final answer then becomes
3x4 − 2x2 − 21 = (3x2 + 7)(p2 − 3)
Write in quadratic form: 5x4 − 6x2 + 1
  • Write the expression in the format ax2 + bx + c
  • 5(x2)2 − 6(x)2 + 1
  • Let p = x2
  • Notice that we can factor 5p2 − 6p + 1
  • ax2 + bx + c = 0
  • ax2 + bx + c = (x + [m/a])(x + [n/a])
  • Find two numbers "m" and "n" such that:
    a*c = m*n b = m + n
  • 5 = m*n
  • − 6 = m + n
  • The only numbers that work is − 5 and − 1
  • 5p2 − 6p + 1 = (p − [5/5])(p − [1/5]) = (p − 1)(5p − 1)
  • Given that p = x2, we get
5x4 − 6x2 + 1 = (x2 − 1)(5x2 − 1)
Write in quadratic form: 16x8 − 17x4 + 1
  • Write the expression in the format ax2 + bx + c
  • 16(x4)2 − 17(x4) + 1
  • Let p = x4
  • Notice that we can factor 16p2 − 17p + 1
  • ax2 + bx + c = 0
  • ax2 + bx + c = (x + [m/a])(x + [n/a])
  • Find two numbers "m" and "n" such that:
    a*c = m*n
    b = m + n
  • 5 = m*n
  • − 6 = m + n
  • The only numbers that work is − 16 and − 1
  • 16p2 − 17p + 1 = (p − [16/16])(p − [1/16]) = (p − 1)(16p − 1)
  • Given that p = x4, we get
  • 16x8 − 17x4 + 1 = (x4 − 1)(16x4 − 1)
  • We repeat the process to factor out (x4 − 1) and (16x4 − 1). Notice how they will will become difference of squares
  • Let q = x2
  • (q2 − 1)(16q2 − 1) = (q2 − 1)((4q)2 − 1) = (q + 1)(q − 1)(4q + 1)(4q − 1)
  • By plugging back in q = x2 we get
16x8 − 17x4 + 1 = (x2 + 1)(x2 − 1)(4x2 + 1)(4x2 − 1)
Solve x4 + 3x2 − 4 = 0
  • Remember that working with x to the 4th power is the same as working with x squared.
  • x2 + bx + c = 0
  • x2 + bx + c = (x + m)(x + n)
  • Find two numbers "m" and "n" such that:
    c = m*n
    b = m + n
  • −4 = m*n
  • 3 = m +n
  • The only two numbers that work is 4 and − 1
  • (x2 + 4)(x2 − 1) = 0
  • Second term is difference of squares, so that will give us.
  • (x2 + 4)(x − 1)(x + 1) = 0
  • Using the Zero Product Property we get
  • x2 + 4 = 0 , x − 1 = 0 , x + 1 = 0
  • Solving the first equation gives us the following imaginary roots
  • 0 = x2 + 4 =
  • x2 = − 4
  • x = ±√{ − 4}
  • = ±i√4 =
  • = ±2i
  • Solution is then
x = 2i
x = − 2i
x = 1
x = − 1
Solve x4 − 6x2 + 8 = 0
  • Remember that working with x to the 4th power is the same as working with x squared.
  • x2 + bx + c = 0
  • x2 + bx + c = (x + m)(x + n)
  • Find two numbers m and n such that:
    c = m*n
    b = m + n
  • 8 = m*n
  • −6 = m + n
  • The only two numbers that work is − 4 and − 2
  • (x2 − 4)(x2 − 2) = 0
  • First term is difference of squares, so that will give us.
  • (x − 2)(x + 2)(x2 − 2) = 0
  • Using the Zero Product Property we get
  • x − 2 = 0 , x + 2 = 0 , x2 − 2 = 0
  • Solving the third equation involves square roots.
  • x2 − 2 = 0
  • x2 = 2
  • x = ±√2
  • Solution is then
x = √2
x = − √2
x = 2
x = − 2
Solve x4 − 2x2 − 63 = 0
  • Remember that working with x to the 4th power is the same as working with x squared.
  • x2 + bx + c = 0
  • x2 + bx + c = (x + m)(x + n)
  • Find two numbers m and n such that:
    c = m*n
    b = m + n
  • −63 = m*n
  • −2 = m + n
  • The only two numbers that work is 7 and − 9
  • (x2 + 7)(x2 − 9) = 0
  • Second term is difference of squares, so that will give us.
  • (x2 + 7)(x − 3)(x + 3) = 0
  • Using the Zero Product Property we get
  • x2 + 7 = 0 , x − 3 = 0 , x + 3 = 0
  • Solving the first equation involves square roots and imaginary numbers.
  • x2 + 7 = 0
  • x2 = − 7
  • x = ±√{ − 7}
  • x = ±i√7
  • Solution is then, solving the remaining two equations
x = i√7
x = − i√7
x = 3
x = − 3
Solve x4 + 8x2 + 16 = 0
  • Remember that working with x to the 4th power is the same as working with x squared.
  • x2 + bx + c = 0
  • x2 + bx + c = (x + m)(x + n)
  • Find two numbers m and n such that:
    c = m*n
    b = m + n
  • 16 = m*n
  • 8 = m + n
  • The only two numbers that work is 4 and 4
  • (x2 + 4)(x2 + 4) = 0
  • This is a special case, whenever the same term repeats, it is called multiplicity 2, if it were to repeat once more,
  • it would be called roots with multiplicity 3
  • (x2 + 4)(x2 + 4) = 0
  • Using the Zero Product Property we get
  • x2 + 4 = 0 , x2 + 4 = 0
  • Solving the first equation involves square roots and imaginary numbers.
  • x2 + 4 = 0
  • x2 = − 4
  • x = ±√{ − 4}
  • x = ±i√4
  • x = ±2i
  • Solution is then,
x = 2i, x = − 2i, x = 2i, and x = − 2i
You can also say:
x = 2i multiplicity 2 and x = − 2i multiplicity 2

*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.

Answer

Solving Polynomial Functions

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
  • Factoring Polynomials 0:06
    • Greatest Common Factor (GCF)
    • Difference of Two Squares
    • Perfect Square Trinomials
    • General Trinomials
    • Grouping
  • Sum and Difference of Two Cubes 6:03
    • Examples: Two Cubes
  • Quadratic Form 8:22
    • Example: Quadratic Form
  • Example 1: Factor Polynomial 12:03
  • Example 2: Factor Polynomial 13:54
  • Example 3: Quadratic Form 15:33
  • Example 4: Solve Polynomial Function 17:24

Transcription: Solving Polynomial Functions

Welcome to Educator.com.0000

In today's lesson, we will be working on solving polynomial equations.0002

And we are going to start out by reviewing some factoring techniques.0006

And the techniques that you will use for factoring polynomials are familiar from earlier work.0010

And these include greatest common factor, difference of two squares, perfect square trinomials, general trinomials, and factoring by grouping.0015

So, just as we did with quadratic equations, if you are going to be working with a polynomial,0025

the first thing you want to do is: if there is a greatest common factor, factor that out first.0031

For example, if I had something like 4x4 - 6x2 - 12x, I have a greatest common factor of 2x.0039

So, I have pulled that out first; and that would give me 2x, leaving behind 2x3; here that would leave behind 3x; and here, -6.0055

And then, I would work on factoring that farther, if it is possible.0069

OK, the difference of two squares: this is the greatest common factor, and now difference of two squares.0074

Recall that these are in the form a2 - b2, and an example would be x2 - 9.0085

This could be factored out into (x + 3) (x - 3); this factors into (a - b) times (a + b).0094

Here, in this case, a equaled x and b equaled 3: x2 - 32.0107

In a few minutes, we will be talking about the difference and sum of two cubes; we will go on a little farther than just working with squares.0118

Next, perfect square trinomials: you know that, if you recognize these, they are easy to work with.0128

So, you want to be on the lookout for these perfect square trinomials.0136

For example, one that we have seen earlier, working with quadratic equations, is x2 + 8x + 16.0140

And this factors as (x + 4)2, or (x + 4) (x + 4)--a perfect square trinomial.0151

And all of these are covered in detail in earlier lectures, so go ahead back and review these, if necessary,0166

to make sure you have them down before you work on factoring polynomials.0173

General trinomials: you would recognize these in a form that they don't fit into the special cases0178

of difference of two squares or perfect square trinomials.0186

It might be something such as, say, x2 + 2x -8; and you need to just use some trial and error on this.0189

For example, for this one, you would say, "OK, the first term in each factor has to be x."0198

And since I have a negative in front of the constant here, I have to have a positive here and a negative here,0203

because a positive times a negative is going to give me a negative.0209

Then, I am going to look at the factors of 8, and I am going to say, "OK, I have 1 and 8, and 2 and 4."0212

And then, I want to find factors of 8 that, when one is positive and one is negative, sum up to 2.0221

And I can see that 1 and 8 are too far apart; they are never going to give me 2.0227

2 and 4 are perfect; and I see that, if I make the 4 positive and the 2 negative,0231

I am going to get 2, which is going to be the coefficient for my middle term.0240

This is going to factor out to (x + 4) (x - 2); and I can always check this by using FOIL,0246

which will tell me that the First term gives me x2; Outer is -2x; Inner is 4x; and then, this is -8.0253

So, x2 + 2x - 8--it gave me this trinomial back.0262

Grouping: recall that we use grouping when we are factoring polynomials with 4 terms--factoring by grouping.0272

Let's say you have some things such as 3x3 - 4x2 + 6x - 8.0282

I am going to handle this by grouping the first two terms, and then grouping the second two terms.0290

Now, I am going to pull out any common factors I see; and I do have a common factor of x2, to leave behind 3x - 1.0299

Over here, I have a common factor of 2; I am going to pull that out, and that is going to leave me with 3x - 4.0311

I am pulling out the x2: that is going to leave me with 3x - 4 here; here I am going to pull out the 2; that is going to leave me with 3x - 4.0321

This is what I wanted to happen; and what I have here is a common binomial factor.0329

I am going to factor that out (pull that out in front), and that leaves behind x2 + 2.0334

So, factoring by grouping: group the first two terms; group the second two; pull out a common factor.0343

If you have a common binomial factor left behind, then you pull that out in front.0349

Make sure you are familiar with all of these; review the ones that you need to.0355

And we are going to be using these to solve polynomial equations.0358

Now, we are going on to a new concept with factoring: and that is the sum and difference of two cubes.0363

You are familiar with working with squares; and now we are talking about the sum and difference of two cubes.0369

Let's look at an example: 64x3 + 125.0375

Well, if I think about this, the cube root of 64 is 4; the cube root of x3 is x; the cube root of 125 is 5.0381

Here, this is in this form, a3 + b3; here, a equals 4x, and b equals 5.0399

Now, by memorizing this formula, I can just factor this out.0409

So, if I know that this is in this form, a3 + b3, which equals0414

(a + b) times (a2 - ab + b2), then I know that I have that a is 4x and b is 5.0420

So, I just have to substitute that in.0430

Here, I am supposed to have a2, so that is going to give me (4x)2, minus a times b, plus b2.0433

OK, this is 4x + 5, and this is going to give me 16x2 - 20x + (52 is) 25.0446

So again, just recognize that this is actually the sum of two cubes, so it is going to factor out to this form.0467

For the difference of two cubes, it is a very similar idea--only you are going to end up with a negative sign here and a positive here.0475

It is the same idea, just different signs.0483

So, 64x3 + 125 factors out to this.0485

And then, from there, in some cases, you may be able to go on with your factoring.0493

Let's talk about quadratic form: we worked with quadratic equations earlier on, and you can actually0502

put higher-order polynomials into a form called quadratic form.0507

And quadratic form is an2 + bn + c, where n is an expression of x.0512

Well, what does that mean--"n is an expression of x?"0520

Let's use an example: let's say I have 6x8 + 5x4 + 9, and I want to get it into this form, an2 + bn + c.0524

Well, let's look at this more closely: I want to get this into some form where I have a number squared, so I have a square here.0540

Now, I am going to think about: "x to the what, squared, equals x8?"0553

Thinking about raising a power to a power, I know that I have to multiply 2 times something to get 8.0558

So, 8 divided by 2 is 4; so that tells me that (x4)2 is x8.0567

So, all I did is took my x8 and put it in a form where it is something squared.0574

Now, I am going to assign another variable this value, x4.0580

I could use y; I could use z; I am going to go ahead and use y--I am going to let y equal x4.0585

Therefore, y2 equals x8, since y equals x4,0596

because if y equals x4, I could just substitute and say that this is really (x4)2.0608

OK, so again, what I did is looked at what I had, x8; and I said,0616

"What would have to be the exponent here, so that I could just square it and get this back?"0621

Well, the exponent would have to be x4.0626

Then, I took this and wrote it as a different variable, y = x4.0628

I assigned the variable y the value x4; so that is what this means--something n2.0634

Here, I am just going to say it is y; so this gives me...my leading coefficient is 6.0640

Now, instead of putting n, I just decided I am going to use y to represent x4, and that is going to be y2.0646

Plus...my next coefficient is 5.0654

OK, here I have an x4, but I said y = x4, so I am just going to write it as y.0657

And then, my constant...so now, I have taken this and rewritten it as this equation, where y is equal to x4.0664

Now that this is in quadratic form, I can use techniques such as the quadratic formula and factoring, learned earlier on,0672

in order to (if this were an equation, let's say, then I could) solve it using these techniques for working with quadratics.0679

Once I find the value of y, whatever y turns out to be, if I know y, I can find x.0688

If y is 16, then I would just have to say, "OK, this equals 2"; then x would equal 2.0696

So, by putting this equation into quadratic form, I can use quadratic techniques to solve for y.0707

Once I have y, I can solve for x, which makes quadratic form very useful when working with certain polynomials.0715

Looking at this first example, 27a3 - 8, I recognize this as the difference of two cubes.0725

And this is in this form; here, the cube root of 27 is 3; the cube root of a3 is a; so this is what I have.0731

The cube root of 8 is 2; therefore, what I have in here is a, and what I have right here is b.0751

And sometimes it helps to actually write these out instead of doing it in your head, just to keep everything straight.0764

Since I know that a is 3a, and b equals 2, I just have to substitute here to factor: a - b--I am going to rewrite it down here:0768

a2 + ab + b2; OK, if a is 3a, then that is going to give me 3a - b (which is 2).0778

Here, I am going to have a2; so that is 3a2 + a times b + b2.0788

This is going to give me 9a2 + 6a + 4.0802

And one important thing is just to watch the signs, working with the difference of two cubes or the sum of two cubes.0810

And then, figure out what your a and b values are by finding the cube root of these terms.0818

And then, just substitute in a and b into this formula to get factorization.0823

In the next example, we are asked to factor this trinomial; and the first step is to factor out a greatest common factor, if there is one.0836

And I am looking here for a greatest common factor.0846

And I see that I have a 3 that can be pulled out of all of these terms.0852

For x's, I have x2; I have an x here; and an x3; so they all have at least an x.0858

For y, there is one y here; there is a y3 here; and there is a y2 here.0865

For z, I have z3, z2, and z4, so they all have at least z2.0871

So, the greatest common factor is 3xyz2; so I am going to pull that out and see what is left behind.0877

In my first term, the 3 is gone; I am left with an x.0885

The y is gone, and the z2 is gone; that leaves me just xz.0889

Here, I have a negative sign; I pulled a 3 out, leaving behind a 4; the x is gone; one y is gone, so I have y2; z2 is gone.0895

Here, I pulled out the factor of 3; that leaves behind 8; one x is gone; I have x2; one y is gone; and z2 is gone.0904

OK, so I am left with the greatest common factor, times what was left behind in here.0915

And I am looking, and there are no common factors left.0921

I don't recognize this as the difference of two squares, or anything that I can factor out.0924

So, this is as far as I can go with the factorization.0929

Now, I am asked to write this polynomial in quadratic form; recall that quadratic form is an2 + bn + c.0935

Here, I have x6; I want to find some variable, assign it a certain value, and then square it.0946

So, x to the something, squared, equals x6.0958

Well, if I take 6 and divide it by 2, I am going to get 3; therefore, (x3)2 = x6, because 3 times 2 is 6.0964

I am going to let y equal x3; therefore, y2 is going to equal x6, because that would be (x3)2.0975

Now, let's look at what we have: we have our leading coefficient of 10, x6...0995

well, x6 is going to be the same as y2; minus 12; now, I have x3 here.1001

Well, I said that y equals x3, so I am just going to put y here; plus 14.1010

This is my given equation (this polynomial) in quadratic form; and again, I did that by assigning y the value of x3,1016

which allowed me to write that as 10y2 - 12y + 14.1024

I could then use methods for solving quadratic equations, if this were an equation.1029

Then, I could solve this and find y; once I have y, I could go back in and just find the cube root of that, which would give me x.1034

In this example, I am actually asked to solve the equation; I am going to start out by factoring.1045

And this is a trinomial; and again, when you see that x4 trinomial, you could think1050

that it is similar to working with a quadratic, except you have x2 here in the factors, instead of just x.1057

So, x2 times x2 is x4.1063

I have a negative sign here, which means that one term is going to be positive, and one is going to be negative (for the second terms).1068

Let's think about the factors of 24, and let's find the factors that are going to add up to -2.1075

Factors of 24: 1 and 24, 2 and 12, 3 and 8, 4 and 6.1084

Now, this is a lot, but I don't have to work with all of these; I want to find ones that are close together, because their sum is just going to be -2.1096

So, the closest together are 4 and 6; and I see that one is positive and one is negative.1103

Well, if I make 4 positive and 6 negative, their sum will be -2.1108

So, I know that this is what I want: I want for 4 to be positive, and I want the 6 to be negative (and this all equals 0).1116

OK, so I have actually factored this as far out as I can; x2 + 4 doesn't factor any farther, nor does this.1126

So, I am just going to go ahead and use the zero product property to solve.1134

And the zero product property tells me that x2 + 4 = 0, or x2 - 6 = 0,1139

because if either of these terms is 0, the product of the two will be 0, and the equation will be solved.1146

Here, I am going to get x2 = -4.1152

Now, when I take the square root of that, I am going to end up with this.1156

Recall from working with imaginary and complex numbers that the square root of -1 equals i.1161

So, this is the same as saying, "The square root of -1 times 4," which equals the square root of -1, times the square root of 4.1171

So, since this equals i, I am just going to write this as this; and that is actually plus or minus;1184

we have to take the positive and the negative, according to the square root property.1191

OK, well, this is a perfect square, so this gives me ± 2i--that is some review of complex numbers.1195

Here, it is a little bit simpler: x2 = 6.1206

According to the square root property, if I take the square root of both sides, then I will get x = ±√6.1209

All right, so I was asked to solve this; and I came up with solutions.1217

And the solutions are that x equals 2i, -2i, √6, or -√6.1220

And I figured that out by factoring this into these two factors, (x2 + 4) times (x2 - 6),1231

and then continuing on to use the zero product property.1243

In this case, I ended up with x2 equaling a negative number, so I ended up having to use a complex number for my result.1250

And then, these are the four solutions to this polynomial equation.1256

That concludes this session of Educator.com on polynomial equations.1262

And I will see you next lesson for more work with polynomials.1267

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