INSTRUCTORS Carleen Eaton Grant Fraser

Home » Mathematics » Algebra 2 » Rational Zero Theorem

Dr. Carleen Eaton

Dr. Carleen Eaton

Rational Zero Theorem

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Table of Contents

Section 1: 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
Section 2: 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
Section 3: 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
Section 4: 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
Section 5: 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
Section 6: 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
Section 7: 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
Section 8: 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
Section 9: 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
Section 10: 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
Section 11: 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 (10)

1 answer

Last reply by: Ben Guo
Mon Jul 13, 2020 11:31 AM

Post by Jerry Jin on July 23, 2019

What if there is no constant?

0 answers

Post by Kenneth Geller on February 18, 2019

Very interesting concepts!! Who in the past figured these somewhat elegant rules out?

1 answer

Last reply by: Dr Carleen Eaton
Fri Jun 8, 2018 11:41 PM

Post by John Stedge on June 7, 2018

What if there is no constant?

1 answer

Last reply by: Dr Carleen Eaton
Fri Jun 8, 2018 11:35 PM

Post by John Stedge on June 7, 2018

In example one is the function supposed to say f(x)=2x^4-3x^2+6x^2-x+9 or f(x)=2x^4-3x^3+6x^2-x+9?

0 answers

Post by julius mogyorossy on December 16, 2013

I was wondering how do you know when you should continue to factor to find the remaining 0's, you say the way you did it in Ex. 4 is simpler, not to me, yet, but I realized, I think, that since the degree is 3, you know if you can only find one positive or real root, that you can, must, factor, to find the remaining roots, the complex roots, is this correct?

1 answer

Last reply by: Dr Carleen Eaton
Thu Jun 13, 2013 11:59 PM

Post by Kavita Agrawal on June 13, 2013

Isn't the constant in the first example, 4x^2 + x - 3, -3 and not 3? I don't think this should make difference, though...

Rational Zero Theorem

  • Understand the Rational Zero Theorem and the special case where the leading coefficient is 1. Use it to list all possible rational roots of a polynomial.
  • Use synthetic substitution to test each possible rational root in your list.
  • After you find the first root, try to factor the quotient to find the remaining roots.

Rational Zero Theorem

List all possible rational zeros x3 + 8 = 0
  • Find [p/q] where p are the factors of the constant and q are the factors of the leading coefficient.
  • p = 8
  • Factors of p: 1,2,4,8
  • q = 1
  • Factors of q: 1
  • [Factors of p/Factors of q] = [1,2,4,8/1]
  • Now Find all the possibilities
[p/q] = ±[1/1]; ±[2/1]; ±[4/1]; ±[8/1] = ±1; ±2; ±4; ±8
List all possible rational zeros 3x4 − 40x3 + 10x2 + 40x − 13 = 0
  • Find [p/q] where p are the factors of the constant and q are the factors of the leading coefficient.
  • p = 13
  • Factors of p: 1,13
  • q = 3
  • Factors of q: 1,3
  • [Factors of p/Factors of q] = [1,13/1,3]
  • Now Find all the possibilities
[p/q] = ±[1/1]; ±[1/3]; ±[13/1]; ±[13/3] = ±1; ±[1/3]; ±13; ±[1/3]
List all possible rational zeros 5x2 + 10x + 2 = 0
  • Find [p/q] where p are the factors of the constant and q are the factors of the leading coefficient.
  • p = 2
  • Factors of p: 1,2
  • q = 5
  • Factors of q: 1,5
  • [Factors of p/Factors of q] = [1,2/1,5]
  • Now Find all the possibilities
[p/q] = ±[1/1]; ±[1/5]; ±[2/1]; ±[2/5] = ±1; ±[1/5]; ±2; ±[2/5]
List all possible rational zeros 5x2 + 4x − 1 = 0
  • Find [p/q] where p are the factors of the constant and q are the factors of the leading coefficient.
  • p = 1
  • Factors of p: 1
  • q = 5
  • Factors of q: 1,5
  • [Factors of p/Factors of q] = [1/1,5]
  • Now Find all the possibilities
[p/q] = ±[1/1]; ±[1/5]; = ±1; ±[1/5]
List all possible rational zeros 9x4 − 4 = 0
  • Find [p/q] where p are the factors of the constant and q are the factors of the leading coefficient.
  • p = 4
  • Factors of p: 1,2,4
  • q = 9
  • Factors of q: 1,3,9
  • [Factors of p/Factors of q] = [1,2,4/1,3,9]
  • Now Find all the possibilities
[p/q] = ±[1/1]; ±[1/3]; ±[1/9]; ±[2/1]; ±[2/3]; ±[2/9]; ±[4/1]; ±[4/3]; ±[4/9] = ±1; ±[1/3]; ±[1/9]; ±2; ±[2/3]; ±[2/9]; ±4; ±[4/3]; ±[4/9]
List all possible rational zeros − 15x4 + 37x3 + 38x2 − 19x − 5 = 0
  • Find [p/q] where p are the factors of the constant and q are the factors of the leading coefficient.
  • p = 5
  • Factors of p: 1,5
  • q = 15
  • Factors of q: 1,3,5,15
  • [Factors of p/Factors of q] = [1,5/1,3,5,15]
  • Now Find all the possibilities
[p/q] = ±[1/1]; ±[1/3]; ±[1/5]; ±[1/15]; ±[5/1]; ±[5/3]; ±[5/5]; ±[5/15] = ±1; ±[1/3]; ±[1/5]; ±[1/15]; ±5; ±[5/3];
Find all the zeros of x3 − 9x2 − 17x + 10 = 0
Find the possible rational zeros
  • [p/q] = [1,2,5,10/1]
  • Possible rational zeros then are
  • ±1; ±2; ±5; ±10
  • Test Using Synthetic Division. Begin by testing ,1, − 1, 2, − 2. If the remainder is Zero, then that number is a rational root.
  • 11-9-1710
      1-8-25
     1-8-25−15
  • -11-9-1710
      -1107
     1-10-717
  • 21-9-1710
      2-14-62
     1-7-31−52
  • -21-9-1710
      -222-10
     1-1150
  • So far x = 2 is a zero, leaving behind x2 − 11x + 5 = 0 to factor out.
  • By observation, you will see that the remaining polynomial is not factorable, the last thing to do is to use the Quadratic Formula to find the missing roots.
Roots = { − 2,[(11 + √{101} )/2],[(11 − √{101} )/2]}
Find all the zeros of 3x3 + x2 − 3x − 1 = 0
  • Find the possible rational zeros
  • [p/q] = [1/1,3]
  • Possible rational zeros then are
  • ±1; ±[1/3]
  • Test Using Synthetic Division. Begin by testing ,1, − 1, [1/3], − [1/3]. If the remainder is Zero, then that number is a rational root.
  • 131-3-1
      341
     3410
  • 131-3-1
      -321
     3-2-10
  • [1/3]31-3-1
      1[2/3]−[7/9]
     32−[7/3]
    −[16/9]
  • −[1/3]31-3-1
      101
     30−3
    0
  • Using synthetic division, you found all possible zeros.
Roots/Zeros = { 1, − 1, − [1/3]}
Find all the zeros of 5x3 − 2x2 − 5x + 2 = 0
  • Find the possible rational zeros
  • [p/q] = [1,2/1,5]
  • Possible rational zeros then are
  • ±1; ±[2/5] ±[1/5]; ±2;
  • Test Using Synthetic Division. Begin by testing ,1, − 1, [2/5], − [2/5]. If the remainder is Zero, then that number is a rational root.
  • 15-2-52
      53-2
     53-20
  • -15-2-52
      -57-2
     5-720
  • [2/5]5-2-52
      [10/5]0−[10/5]
     50-50
  • −[2/5]5-2-52
      −[10/5][8/5][34/25]
     5-4−[17/5]
    [84/25]
  • Using synthetic division, you found all possible zeros.
Roots = { 1, − 1, − [2/5]}
Find all the zeros of 2x3 + 5x2 + x − 2 = 0
  • Find the possible rational zeros
  • [p/q] = [1,2/1,2]
  • Possible rational zeros then are
  • ±1; ±2; ±[1/2];
  • Test Using Synthetic Division. Begin by testing , − 1, − 2, − [1/2]. If the remainder is Zero, then that number is a rational root.
  • 1251-2
      -2-32
     23-20
  • -2251-2
      -4-22
     21-10
  • −[1/2]251-2
      -1-2[1/2]
     24-1
    −[3/2]
  • Using synthetic division, you found two zeros, x = − 1;x = − 2. There is one more missing.
  • If you looked carefuly when we tested x = − [1/2], it looked like it was going to work. Test x = [1/2]
  • to see if that will work.
  • [1/2]251-2
      132
     264
    0
  • It worked! You have now found all three zeros/roots.
Roots = { − 1, − 2,[1/2]}

*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

Rational Zero Theorem

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
  • Equation 0:08
    • List of Possibilities
    • Equation with Constant and Leading Coefficient
    • Example: Rational Zero
  • Leading Coefficient Equal to One 7:19
    • Equation with Leading Coefficient of One
    • Example: Coefficient Equal to 1
  • Finding Rational Zeros 12:58
    • Division with Remainder Zero
  • 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

Transcription: Rational Zero Theorem

Welcome to Educator.com.0000

In today's lesson, we will be discussing the rational zero theorem.0002

What this theorem tells you is the possible rational zeroes of a polynomial.0009

And again, it just helps you generate a list of possibilities.0017

So, if f(x) is some polynomial function with integer coefficients, then you can generate a list of possible rational zeroes0020

by looking at two things: the leading coefficient and the constant.0032

If p/q is a fraction in simplest form that is a zero of f(x), then p is a factor of a0,0042

which is the constant; and q is a factor of an, which is the leading coefficient.0053

So, let's think about what this is saying.0061

p/q: here, p is a factor of the constant; so the numerators of all the elements of the lists that we are going to generate consist of factors of the constant.0069

The denominators consist of factors of the leading coefficient.0091

Using this equation, using the rational zero theorem, what I can do is make a list of the factors of the constant,0105

make a list of the factors of the leading coefficient, and then find all possible combinations,0116

using those factors as the numerator (the factors of the constant) and the factors of the leading coefficient as the denominator.0124

Find all the possible combinations; and then, that will give me a list of possible rational zeroes.0133

Now, this is only a list; and we are going to talk about how you check the list and determine which ones,0138

if any, are actually zeroes; because it may turn out that none of these are zeroes.0145

It may turn out that the zeroes are irrational or complex numbers.0150

But if there are rational zeroes, they will be members of this list.0159

OK, using an example: f(x) = 4x2 + x - 3.0165

Here, my leading coefficient is 4, and the constant is 3.0175

Now, I am going to first just think about the factors: the factors of 4 are going to be 1, 2, and 4.0190

And these could be positive or negative; and I am going to worry about the signs in a minute.0208

Right now, I am just going to look for the factors.0212

The factors of 3 are 1 and 3; now, these factors of the leading coefficient are going to comprise the denominator.0219

So, these are possibilities for the denominator.0230

Here, these factors of the constant are going to be possibilities for the numerator.0237

So, I am going to take each one; I am going to go in an organized manner;0246

I am looking for different possibilities for p/q, for the factors of 3 over the factors of 4.0250

p/q could be...I could have 1 in the numerator and 1 in the denominator, and it could be positive or negative,0257

because the factors here could include -1 times -4 (that would give me 4), and so on.0267

So, I could have all different combinations of positive and negative.0274

OK, I could have 1 for the numerator and 2 for the denominator, plus or minus.0277

I could have 1/4; I could have 3/1, plus or minus; I could have 3/2, plus or minus; and plus or minus 3/4.0286

And then, I am just writing this in simpler form: this is just ±1, ±1/2, ±1/4; this is just 3 (3/1 is just 3); ± 3/2; ±3/4.0306

So, this gives me the list of possible rational zeroes.0328

Now, let's factor this out and see what happens, just to check and make sure that what we come up with actually is on this list.0341

Let's take the corresponding equation and find the zeroes.0348

4x2 + x - 3 = 0: you can actually factor this out, and this is going to come out to (4x - 3) (x + 1).0352

So, this would be 4x2, and this would be 4x - 3x (would give me x); -3 times 1 is -3.0367

Using the zero product property, I am going to get 4x - 3 = 0 and x + 1 = 0.0377

So, this is going to give me 4x = 3, or x = 3/4; this will give me x = -1.0385

I look, and I have 3/4 on the list, and I have -1 on the list; both are on the list.0395

So, what this gave me is a list of 1, 2, 3, 4, 5, 6 possibilities, positive or negative: that is 12 possibilities.0402

And I found two of them; I was able to find this by factoring.0410

But what it is showing is that these were on the list; but again, this is just a list of possibilities.0415

And you need to be able to figure out which possibilities are correct.0422

So, what we are going to do is generate lists of possibilities, and then talk about0426

what you can do with that information--how you can find which possibilities are correct.0433

OK, if the leading coefficient of the polynomial is equal to 1, then the rational zero theorem tells us that any rational zero of f(x) is a factor of the constant.0438

Let's think about what this means: we said that the list of rational zeros equal p/q,0454

where p is factors of the constant--let's just say the constant--and q are factors of the leading coefficient.0465

Well, if the leading coefficient is 1, then this becomes p/1, so it is just going to be factors of the constant.0487

So, in the situation where the leading coefficient is 1, you just have to look at the factors of the constant.0515

And those will give you your list of possible rational zeroes.0520

For example, if I have f(x) = x4 - 3x3 + 2x2 + x - 8.0526

I have a leading coefficient of 1, so I don't need to worry about that: p/q = p/1 = p.0543

Now, here p is going to equal the factors of the constant, which will be factors of 8, equal ±1, ±2, ±4, ±8.0553

Now, I mentioned that these are just possibilities; so I need to check these.0576

And if you will recall, we can check to see if something is a factor by using synthetic division.0581

We also talked about how, if a number is a 0, then if you take the value of the function,0592

you are going to find that if f(a) = 0, then x - a is a factor.0605

So, for example, if x - 2 is a factor of f(x), then f(2) will equal 0.0615

And recall that using synthetic division, we can find the value of a function by looking at the remainder.0635

So, what we can do is go back to synthetic division (or synthetic substitution, as we call it in this case)0641

and divide the polynomial by x - a, where a is one of the factors we are checking.0646

And if the remainder is 0, then we know we have a factor or a rational zero, in this case, if the remainder is 0.0653

So, these factors are also zeroes, because f(a) is 0.0664

So, if I go ahead and divide, for example, f(x) by x - 2, that will allow me to check to see if 2 is a factor.0669

If 2 is a factor, the remainder will equal 0.0683

Let's go ahead and do that: here we are dividing by 2, and I do not have any missing terms.0690

I have a coefficient of 1, a coefficient of -3, a coefficient of 2, another coefficient of 1, and a coefficient of -8.0697

Bring down 1; multiply 1 by 2; that gives me 2.0708

And now, I take 2 plus -3 to get -1; multiply -1 times 2 to get -2.0714

Combining 2 and -2, I get 0; 0 times 2 is 0; 1 + 0 is 1; 1 times 2 is 2, to give me a final value of -6; this is the remainder.0722

2 is not a zero of f(x); if the remainder here was 0, then this would tell me that 2 is a zero of f(x),0745

because it is telling me that f(2) equals 0; in other words, when x = 2, the function equals 0.0760

And by definition, that is a zero; it is crossing the x-intercept.0767

OK, to sum up: if you are looking for rational zeroes, list all possible rational zeroes using the theorem.0772

And the theorem says that p/q equals the factors of the constant, over the factors of the leading coefficient.0783

Once you have done that, use synthetic division to determine which possibilities are actually zeroes.0803

Division will yield a remainder equal to 0, if the value is a zero.0812

So, you get your values for p and q; you take the function, and you divide it by each of the values that you are trying, p/q.0832

And if the remainder equals 0, p/q is a zero.0844

If the remainder is anything other than 0, then it is not.0854

So, list the rational zeroes, and then use synthetic division to check.0856

OK, list all possible rational zeroes of this function.0862

I need to find values of p/q, and I am going to first look at the constant; this is a0 = 9, and the factors are 1, 3, and 9.0866

So, these are values for p.0887

Here, the leading coefficient equals 2; factors are just 1 and 2.0893

So now, I am going to go through all of these different combinations of numerator (1, 3, 9) and denominator (1 and 2).0903

So, possible rational zeroes are ±1/1 (I am just going to simplify that to 1 right away), or ±1/2.0911

Next is ±3/1 (which is just 3); ±3/2; next, ±9/1 (which gives me 9); ±9/2.0922

So, I have 1, 2, 3 times 2; that is 6 possibilities; and then plus or minus for each gives me 12 possible rational zeroes.0944

I don't know which, if any, of these are actually zeroes; I would have to check using synthetic division.0956

In this next example, we are asked to actually find the rational zeroes, not just to list the possibilities.0963

So, I am going to need to use synthetic division.0969

Since the leading coefficient is 1, then possible rational zeroes equal the factors of 3.0971

I don't need to worry about this; the denominator is just going to be 1.0986

OK, factors of 3 are 1 and 3; so my possibilities are ±1 and ±3.0990

Now, I need to check these using synthetic division.1003

So, I am starting out with 1; and I am looking, and there are no missing terms: x3, x2, x, and a constant.1006

So, I can just go ahead and put these coefficients: 1, -2, -2, -2, -3.1015

This is 1 times 1 is 1; combining that with -2 gets -1; 1 times -1 is -1, plus -2 is -3; -3 times 1 is -3; this is -6.1022

The remainder equals -6, so 1 is not a zero.1043

OK, now let's try -1 right here: again, my coefficients are the same: 1, -2, -2, -3.1050

OK, this is going to give me 1 times -1 (is -1); -2 and -1 is -3; -3 times -1 is 3.1065

I am combining that with -2 to get 1; times -1 gives me -1 and -3; this is -4; so this is not a zero.1080

OK, the next possibility: let's try 3; the coefficients are 1, -2, -2, -3.1094

This is going to give me 1; times 3 is 3; 3 and -2 is going to give me 1.1105

1 times 3 is going to give me 3 again; and -2 is going to give me 1.1115

1 times 3 is 3; 0; 3 is a zero, because the remainder equals 0; 3 is a zero.1125

OK, I found one rational zero: I have one more possibility left--I checked 1; I checked -1; I checked 3; now I need to check -3.1138

1, -2, -2, -3: bring down the 1 and multiply by -3 to get -3.1150

Combine with -2 to give me -5; -5 times -3 is 15; plus -2 leaves me with 13; times -3 is -39.1158

And that is -42, so this is not a zero.1174

The rational zero for this function is 3; 3 is a rational zero.1181

I had this list of possibilities, and the only one that turned out to be correct, an actual zero, is 3.1188

List all possible rational zeroes of this function.1199

So, I am going to be looking for p/q, which equals the factors of the constant over factors of the leading coefficient.1202

What this is going to give me is 09 1, 3, 5, and 15, over factors of 4: 1, 2, and 4.1228

So, I need to figure out all of these possibilities and use positive and negative for each.1251

So, 1/1 is ±1; and then, 1/2 is ±1/2; 1/4 is ±1/4.1255

OK, now going on to 3: 3/1 is just ±3; 3/2 is ±3/2; 3/4 is ±3/4.1269

OK, next, 5: 5/1 is just 5; 5/2; 5/4...1285

Then, 09 15/1--just 15; 15/2; and then 15/4, plus or minus.1300

So, there are a lot of possibilities here; and I listed them all out.1316

But if you were actually asked to find the rational zeroes, you would need to check these through synthetic division.1320

OK, here we are asked to find all zeroes of this function; and notice, it doesn't just say "rational zeroes"; it says "all zeroes."1325

But I am going to start out by looking at the rational zeroes.1331

And since the leading coefficient is 1, the possible rational zeroes will equal factors of the constant.1334

And that is because, if I am taking p/q, and the denominator is the factors of the leading coefficient...1349

here the only factor is 1; so I am just going to end up with p.1357

So, factors of 4 are ±1, ±2, and ±4.1362

Now, I have a list of possible rational zeroes, and I am going to use synthetic division to determine which of these are actual zeroes (not just possibilities).1374

The coefficient here is 1; the coefficient here is 2; -2; and -4.1387

Now, there are no missing terms: I have cubed, squared, x, and constant.1394

So, I am going to go ahead and use synthetic division; this is 1 times 1, is 1.1399

So, this gives me 2 and 1, which is 3; 3 times 1 gives me 3; that combined with -2 is 1.1406

1 times 1 is 1, combined with -4 is -3.1418

This is not a zero: 1 is not a zero, because the remainder is something other than 0.1423

Let's try -1: the coefficients are, again, 1, 2, -2, and -4.1428

Here, I have 1 times -1 to yield -1; combined with 2 is going to give me 1; this times -1 is -1.1439

Combined with -2, it is -3; -3 times -1 is 3; plus -4 is -1; again, this is not a zero.1453

So, not a zero, not a zero...let's try 2.1467

OK, again, same coefficients, same process: bring down the 1; multiply by 2; combine to get 4.1472

4 times 2 is 8; plus -2 leaves me with 6; times 2 is 12; plus -4 is 8--again, not a zero.1488

Now, I am going to try -2: bring down the 1; times -2; combining 2 and -2 leaves me 0, times -2...1502

Combining, you get -2 times -2, is positive 4; -4 and positive 4 is 0.1520

So, looking at this, since the remainder equals 0, -2 is a zero of this function.1530

Now, I could continue on to check these two; but there is an easier way to proceed.1538

because what this is telling me is that (remember, this is the opposite sign) x + 2 (because I took the opposite sign) is a factor of this.1545

So, that leaves me with x + 2, times...this is x3, so times 1x2, plus 0x (that drops out), minus 2.1563

This is what I am actually left with: (x + 2) (x2 - 2).1575

So, instead of just checking these two (and they might be right; they might not be right),1581

what I am going to need to do, if these are not correct (because there are three zeroes, because the degree is 3)1588

is: then I am going to need to find irrational or complex zeroes.1595

So, instead of bothering to even check these, I just use the fact that I now have (x2 - 2) (x + 2) to go ahead and use factoring to solve.1600

So, I know that I have x + 2, and I look, and I can't factor any more.1613

So, I use the zero product property; and I know already that x + 2 = 0, and x = -2; I figured that out.1621

But I can go on and do it with this, as well: x - 2 = 0.1630

And this is going to give me x2 = 2; I am then going to take the square root of both sides to get x = ±√2.1634

I have two zeroes here, and I have one here: the zeroes are -2, √2, and -√2.1645

And this is an irrational number, so it wasn't on my list of possibilities.1656

It is easier to proceed by finding one zero and then using the information that you have here to write the factored-out1660

(or at least partially-factored-out, in some cases) form of the equation,1668

and then go on and use the zero product property to find the remaining zeroes.1672

Thanks for visiting Educator.com, and I will see you next lesson!1678

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