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INSTRUCTORS Carleen Eaton Grant Fraser
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Dr. Carleen Eaton

Dr. Carleen Eaton

Roots and Zeros

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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 (2)

1 answer

Last reply by: Jerry Xu
Fri Aug 3, 2018 1:44 PM

Post by julius mogyorossy on December 16, 2013

x=+-2i, so what that is really saying is, x=2, after you square it, make it negative, is that it?

Roots and Zeros

  • A polynomial of degree n has n roots.
  • If the coefficients of the polynomial are real, then complex roots occur as conjugate pairs.
  • Learn and understand Descartes’ rule of signs for the roots of a polynomial.
  • Use synthetic substitution to find the roots of a polynomial of degree 3 or more.

Roots and Zeros

Solve x3 − 5x2 + 6x = 0
  • Find the GCF
  • GCF = x
  • x(x2 − 5x + 6) = 0
  • Find two numbers such that when multiplied = 6; and when added = − 5
  • Those two numbers are − 3 and − 2
  • x(x − 3)(x − 2) = 0
  • Using Zero Product Property we have
  • x = 0, x − 3 = 0, and x − 2 = 0
  • Solve
x = 0, x = 3, and x = 2
Solve x3 − 3x2 − 4x = 0
  • Find the GCF
  • GCF = x
  • x(x2 − 3x − 4) = 0
  • Find two numbers such that when multiplied = − 4; and when added = − 3
  • Those two numbers are − 4 and 1
  • x(x − 4)(x + 1) = 0
  • Using Zero Product Property we have
  • x = 0, x − 4 = 0, and x + 1 = 0
  • Solve
x = 0, x = 4, and x = − 1
Solve 2x3 − 2x2 − 40x = 0
  • Find the GCF
  • GCF = 2x
  • 2x(x2 − x − 20) = 0
  • Find two numbers such that when multiplied = − 20; and when added = − 1
  • Those two numbers are 4 and − 5
  • 2x(x + 4)(x − 5) = 0
  • Using Zero Product Property we have
  • 2x = 0, x + 4 = 0, and x − 5 = 0
  • Solve
x = 0, x = − 4, and x = 5
Solve x4 − 17x2 + 16 = 0
  • Recall that factoring a 4th power polynomial is the same as factoring a 2nd degree polynomial.
  • Find two numbers m and n such that when multiplied, m*n = 16 and when added m + n = − 17
  • Those numbers are − 16 and − 1
  • (x2 − 16)(x2 − 1) = 0
  • Notice that nowe have two difference of squares.
  • (x − 4)(x + 4)(x − 1)(x + 1) = 0
  • Solve using Zero Product Property
  • x − 4 = 0, x + 4 = 0, x − 1 = 0, and x + 1 = 0
  • Solve
x = 4, x = − 4, x = 1, and x = − 1
Solve x4 − 17x2 + 16 = 0
  • Recall that factoring a 4th power polynomial is the same as factoring a 2nd degree polynomial.
  • Find two numbers m and n such that when multiplied, m*n = 16 and when added m + n = − 17
  • Those numbers are − 16 and − 1
  • (x2 − 16)(x2 − 1) = 0
  • Notice that nowe have two difference of squares.
  • (x − 4)(x + 4)(x − 1)(x + 1) = 0
  • Solve using Zero Product Property
  • x − 4 = 0, x + 4 = 0, x − 1 = 0, and x + 1 = 0
  • Solve
x = 4, x = − 4, x = 1, and x = − 1
Solve x4 − 3x2 − 54 = 0
  • Recall that factoring a 4th power polynomial is the same as factoring a 2nd degree polynomial.
  • Find two numbers m and n such that when multiplied, m*n = − 54 and when added m + n = − 3
  • Those numbers are 6 and − 9
  • (x2 + 6)(x2 − 9) = 0
  • Notice that we have one difference of squares.
  • (x2 + 6)(x − 3)(x + 3) = 0
  • Solve using Zero Product Property
  • x2 + 6 = 0, x − 3 = 0, and x + 3 = 0
  • Solve
x = ±i√6 , x = 3, x = − 3
Determine the possible combinations of possible real roots, negative real roots and complex roots.
3x5 − 15x4 − 29x3 + 145x2 + 40x − 200 = 0
  • Type of Root: Total Roots
    How to Find It: You will find the total number of roots by the Degree of the Polynomial
  • Type of Root: Total Positive Real Roots
    How to Find It: Find the number of sign changes or is less than this by an even number. If number of sign changes is 5, then there could be, 5, 3, 1 positive real roots.
  • Type of Root: Total Negative Real Roots
    How to Find It: Compute f(-x). Count the number of sign changes, or is less than this by an even number.
  • Type of root: Complex Roots
    How to Find It: ComplexRoots = Total Roots - Positve Real Roots - Negative Real Roots
  • What is the total possible number of roots?
  • Total Possible roots is 5 because Degree is 5.
  • How Many Positive Real Root? How many Sign Changes are there?
  • According to Descarte's Change of Signs Rule there could be 3 Positive Real Roots, or less than an even number
  • Positive Real Roots = 3 or 1
  • How many Negative Real Roots are there? Compute f( − x) then count the number of sign changes
  • f( − x) = 3( − x)5 − 15( − x)4 − 29( − x)3 + 145( − x)2 + 40( − x) − 200
  • f( − x) = − 3(x)5 − 15x4 + 29x3 + 145x2 − 40x − 200
  • There couuld be 2 Negative Real Roots because there were only 2 sign changes, or less than that by an even number.
  • Negative Real Roots, 2 or 0
  • Complex Roots can be found by adding the different combinations of + Real and - Real Roots as follows
These are all the possible combinations of positive real, negative real and complex roots:

+Real-RealComplexTotal
3205
3025
1225
1045
Determine the possible combinations of possible real roots, negative real roots and complex roots.
2x5 + 10x4 − 3x3 − 15x2 − 35x − 175 = 0
  • Type of Root: Total Roots
    How to Find It: You will find the total number of roots by the Degree of the Polynomial
  • Type of Root: Total Positive Real Roots
    How to Find It: Find the number of sign changes or is less than this by an even number. If number of sign changes is 5, then there could be, 5, 3, 1 positive real roots.
  • Type of Root: Total Negative Real Roots
    How to Find It: Compute f(-x). Count the number of sign changes, or is less than this by an even number.
  • Type of root: Complex Roots
    How to Find It: ComplexRoots = Total Roots - Positve Real Roots - Negative Real Roots
  • What is the total possible number of roots?
  • Total Possible roots is 5 because Degree is 5.
  • How Many Positive Real Root? How many Sign Changes are there?
  • According to Descarte's Change of Signs Rule there could be 1 Positive Real Roots, or less than an even number
  • Positive Real Roots = 1
  • How many Negative Real Roots are there? Compute f( − x) then count the number of sign changes
  • f( − x) = 2( − x)5 + 10( − x)4 − 3( − x)3 − 15( − x)2 − 35( − x) − 175
  • f( − x) = − 2(x)5 + 10(x)4 + 3(x)3 − 15(x)2 + 35(x) − 175
  • There could be 4 Negative Real Roots because there were only 4 sign changes, or less than that by an even number.
  • Negative Real Roots, 4, 2 or 0
  • Complex Roots can be found by adding the different combinations of + Real and - Real Roots as follows
These are all the possible combinations of positive real, negative real and complex roots:

+Real-RealComplexTotal
1405
1225
1045
Determine the possible combinations of possible real roots, negative real roots and complex roots.
10x5 − 4x4 + 5x3 − 2x2 − 50x + 20 = 0
  • Type of Root: Total Roots
    How to Find It: You will find the total number of roots by the Degree of the Polynomial
  • Type of Root: Total Positive Real Roots
    How to Find It: Find the number of sign changes or is less than this by an even number. If number of sign changes is 5, then there could be, 5, 3, 1 positive real roots.
  • Type of Root: Total Negative Real Roots
    How to Find It: Compute f(-x). Count the number of sign changes, or is less than this by an even number.
  • Type of root: Complex Roots
    How to Find It: ComplexRoots = Total Roots - Positve Real Roots - Negative Real Roots
  • What is the total possible number of roots?
  • Total Possible roots is 5 because Degree is 5.
  • How Many Positive Real Root? How many Sign Changes are there?
  • + 10x5 − 4x4 + 5x3 − 2x2 − 50x + 20 = 0
  • According to Descarte's Change of Signs Rule there could be 4 Positive Real Roots, or less than an even number
  • Positive Real Roots = 4 or 2 or 0
  • How many Negative Real Roots are there? Compute f( − x) then count the number of sign changes
  • f( − x) = 10( − x)5 − 4( − x)4 + 5( − x)3 − 2( − x)2 − 50( − x) + 20
  • f( − x) = − 10(x)5 − 4(x)4 − 5(x)3 − 2(x)2 + 50(x) + 20
  • There could be 1 Negative Real Roots because there were only 1 sign changes, or less than that by an even number.
  • Negative Real Roots: 1
  • Complex Roots can be found by adding the different combinations of + Real and − Real Roots as follows
These are all the possible combinations of positive real, negative real and complex roots:

+Real-RealComplexTotal
4105
2125
0145
Determine the possible combinations of possible real roots, negative real roots and complex roots.
64x6 − 1 = 0
  • Type of Root: Total Roots
    How to Find It: You will find the total number of roots by the Degree of the Polynomial
  • Type of Root: Total Positive Real Roots
    How to Find It: Find the number of sign changes or is less than this by an even number. If number of sign changes is 5, then there could be, 5, 3, 1 positive real roots.
  • Type of Root: Total Negative Real Roots
    How to Find It: Compute f(-x). Count the number of sign changes, or is less than this by an even number.
  • Type of root: Complex Roots
    How to Find It: ComplexRoots = Total Roots - Positve Real Roots - Negative Real Roots
  • What is the total possible number of roots?
  • Total Possible roots is 6 because Degree is 6.
  • How Many Positive Real Root? How many Sign Changes are there?
  • + 64x6 − 1 = 0
  • According to Descarte's Change of Signs Rule there could be 1 Positive Real Roots, or less than an even number
  • Positive Real Roots = 1
  • How many Negative Real Roots are there? Compute f( − x) then count the number of sign changes.
  • f( − x) = 64( − x)6 − 1
  • f( − x) = 64x6 − 1
  • There could be 1 Negative Real Roots because there were only 1 sign changes, or less than that by an even number.
  • Negative Real Roots: 1
  • Complex Roots can be found by adding the different combinations of + Real and − Real Roots as follows
These are all the possible combinations of positive real, negative real and complex roots:

+Real-RealComplexTotal
1146

*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

Roots and Zeros

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
  • Number of Roots 0:08
    • Not Nature of Roots
    • Example: Real and Complex Roots
  • Descartes' Rule of Signs 2:05
    • Positive Real Roots
    • Example: Positve
    • Negative Real Roots
    • Example: Negative
  • Finding the Roots 9:59
    • Example: Combination of Real and Complex
  • Conjugate Roots 13:18
    • Example: Conjugate Roots
  • Example 1: Solve Polynomial 16:03
  • Example 2: Solve Polynomial 18:36
  • Example 3: Possible Combinations 23:13
  • Example 4: Possible Combinations 27:11

Transcription: Roots and Zeros

Welcome to Educator.com.0000

We are going to continue our lesson on polynomials by discussing roots and zeroes.0002

The number of roots can be determined by looking at the degree of the polynomial equation,0008

because a polynomial equation of degree n has n roots.0014

This tells you only the number of roots; it doesn't tell you about the nature of the roots; the roots may be real or complex.0018

For example, if I am given a polynomial such as 5x6 + 2x3 - x2 + 9x - 2 = 0,0025

a polynomial equation, I see that the degree equals 6; this tells me that there are 6 roots for this equation.0038

Now, they can be real; and for example, these roots could be something like 1, -2...those are rational numbers.0053

They actually could be irrational numbers, like the square root of 3.0065

You also can have complex numbers as roots.0069

Recall that complex numbers have two parts--something like 2 + 3i; and the first part here is real, and the second part is imaginary.0073

And these are in the form a + bi.0086

We are going to talk, in a few minutes, about how these complex roots occur as conjugate pairs.0091

For example, 2 + 3i and 2 - 3i would be a conjugate pair--the same values here, but opposite signs; or 4 + i and 4 - i.0097

So, I know that here I have 6 roots; but I have no idea if they are real, rational, irrational, or complex, or what combination of those.0111

However, I can determine at least some of that through using Descartes' Rule of Signs.0119

And there are two sections to this rule: the first section helps you determine the possible numbers of positive real roots.0125

Then, we will talk about determining the possible number of negative real roots.0136

So, first we are just looking at the positive real roots: if we let p(x) be a polynomial with real coefficients (no imaginary coefficients--0141

just real coefficients), arranging in descending powers, recall that descending powers would be something like0154

f(x) = x6 + 5x5 + 7x4 - x3 + 6x2 - 2x + 4.0164

This is a degree 6 polynomial; and I have 6, 5, 4, and on down.0180

If you are trying to work with Descartes' Rule of Signs, the first thing to do is check the polynomial.0187

And if it is not arranged in descending powers, you need to put it that way.0192

OK, so I have this arranged in descending powers; the number of positive real roots0196

is the number of changes in sign of the coefficients, or is less than this by an even number.0202

Let's look at what that means: I want to look for the number of sign changes of the coefficient.0209

So, here I have...this coefficient is a 1, and that is positive; 5 is positive; 7 is positive; -1--so there is a sign change.0217

This is positive; this is negative; so that is one sign change.0230

OK, now here, I am going from -1 to positive 6; again, I have a sign change.0237

Here, I am going from a positive to a negative--another sign change; negative to positive--another sign change.0245

That gives me 1, 2, 3, 4--4 sign changes means there are 4 positive real roots, or less than this by an even number.0254

So, less than 4 by an even number would mean 4 - 2 (would give me 2), or another even number, 4: 4 - 4 would give me 0.0275

There are 4, 2, or 0 positive real roots.0287

So again, take the polynomial; arrange it in descending powers; and then look at the number of sign changes.0295

I have 1, 2, 3, 4 sign changes; that tells me that I will have, at most, four positive real roots.0302

However, I may have less than this by an even number (4 - 2: 2; 4 - 4; 0).0312

So, I have three possibilities: I may have 4 positive real roots, 2 positive real roots, or 0 real roots.0319

And I have, since the degree equals 6, a total of 6 roots.0327

I have a total of 6; of these, 4, 2, or 0 may be positive real roots.0338

OK, the second part of this is looking at the number of negative real roots.0343

The number of negative real roots is the number of changes in sign of the coefficients of the terms p(-x), or is less than this by an even number.0350

Let's continue on with the example that we just looked at, where we were given0361

f(x) = x6 + 5x5 + 7x4 - x3 + 6x2 - 2x + 4.0366

Now, I found there were four sign changes, which means 4, 2, or 0 positive real roots.0383

For the negative real roots, I have to look at f(-x).0388

So, I need to change these x's to -x and be very careful with the signs; OK.0394

So, this is going to give me a coefficient here of -1.0416

But if I take a negative to an even power, it is going to become positive.0421

-1 to the sixth power is going to become positive, so this is going to give me f(-x); here it is just going to be x6.0425

Here, I have -1 times x; you can look at it that way--it is -1 times x5.0435

Well, if I take a -1 to an odd power, it is going to remain negative; so, -x5 times 5 is going to give me -5x5.0443

Here, I have a negative coefficient to an even power; it is going to become positive, so this is really 7x4.0457

Here, I have a negative coefficient to an odd power, so it will remain negative; so this is -x3,0467

but it is times a negative: so -x3 times -1 becomes + x3.0475

OK, I have a -x; this actually should be outside...(-x)2 is going to give me -x times -x; that is going to give me a positive.0485

So, this is going to be plus 6x2.0501

Here, I have -x times -2; that is +2x; and my constant remains positive.0507

OK, number of sign changes: I am going to look for the changes in sign.0516

This is positive out here; so a positive to a negative--that is 1; a negative to a positive--that is 2.0523

This stays positive, positive, positive, positive; the number of sign changes equals 2.0533

OK, so the number of negative real roots is the number of changes in sign of the coefficients of the term p(-x), or less than this by an even number.0539

Therefore, I am going to have two negative real roots, or less than this by an even number.0552

Well, 2 minus 2 is 0; I can't go any lower than that for the number of roots; so there are 2 or 0 negative real roots.0559

Again, this power is 6; I have 6 total roots; I have 2 or 0 negative real roots; and last slide, we talked about having 4, 2, or 0 positive real roots.0575

So, we covered the real roots; there also may be complex roots, so let's talk now about the total roots and the different combinations that you could have.0589

We can use Descartes' Rule of Signs to determine the possible combinations of the real and complex roots.0600

So, in that example above, f(x) = x6 + 5x5 + 7x4 - x3 + 6x2 - 2x + 4,0607

the total roots (this is real and complex) equal 6.0624

Positive real roots: using Descartes' Rule of Signs, I found that I could have 4, 2, or 0.0632

Negative real roots: I found f(-x) and looked for the sign changes, and found two of those.0645

So, the number of negative real roots could be 2 or 0.0650

Now, I can figure out the combinations: I know that they need to total 6, and I have0661

my positive real roots' possibilities and my negative real roots; and the last part of this is complex roots.0669

So, I have positive real roots, the number of negative, and the number of complex; and these need to total 6; the total must equal 6.0681

So, I said that, for positive real roots, I could have 4; then, I may have 2 negative real roots.0693

4 and 2 is 6, so that leaves me with no complex.0700

I could have 4 positive real roots; I could have 0 negative real roots; 4 and 0 is 4; to total 6, I will have to have 2 complex roots.0703

OK, another possibility: I have 2 positive real roots and 2 negative real roots: 2 and 2 is 4; 2 more complex roots will give me 6.0717

Or I could have 2 positive real roots and 0 negative real roots.0731

2 and 0 is 2; to total 6, I will have 4 complex roots.0737

OK, finally, I may have 0 positive real roots and 2 negative real roots.0744

0 and 2 is 2, so I have to have 4 complex roots.0753

Then, I may have 0 positive real roots and 0 negative real roots, and that gives me 0 and 0.0760

So, to total 6, I would have to have 6 right here.0774

And since 3 times 2 is 6, I expect 6 combinations, and that is what I have here.0780

So, Descartes' Rule of Signs, and knowing that the degree is 6 (so I have 6 total roots)0785

allows me to figure out the possible combinations of the roots of a polynomial.0790

Conjugate roots: we talked a bit about complex conjugates and the fact that there are complex roots.0798

But getting into a bit more detail: if the polynomial p(x) has real coefficients...0806

the coefficients are not imaginary, which is what we will be working with--real coefficients--for now...0812

the complex roots of p(x) occur as complex conjugates.0819

And you saw this earlier on, when we worked with quadratic equations and quadratic functions.0823

Thinking about something like this: x2 + 4 = 0, if I wanted to find the solutions for this, I could say, "OK, x2 = -4."0830

Now, using the square root property, I take the square root of both sides; and this gives me x = ±√-4.0851

Recall that the square root of -1 equals i; then I can rewrite this as x = ±√-1, times √4,0861

or x = ±i√4, or x equals ±2i.0875

And we can look at this, also, in a different way: we can say x = 0 + 2i, or x = 0 - 2i.0886

And you can then see that this is a pair of complex conjugates.0896

And it is because of this, where we are finding the square root, that we end up with plus or minus some number.0902

And therefore, these complex roots of polynomials occur as complex conjugates.0909

Another example of complex conjugates would be, say, 8 + 5i and 8 - 5i--the same values, but switch the sign--or 2 + 7i and 2 - 7i.0918

And this explains why, when we talked about the number of negative real roots and positive real roots,0935

we said it was that number (say, 4), or less than that by 2.0943

And the reason it goes down by pairs is because the complex conjugates occur in pairs.0946

So, they could take up two of the spots for the roots; and so they are going to decrease the number of the real roots by 2 or multiples of 2.0951

Here, we are asked to solve this polynomial equation, x3 - x2 - 6x = 0.0965

I am going to solve this by factoring; and the first step is to factor out the greatest common factor.0975

And here, I see I have a common factor of x; there is an x in each of these.0980

So, factor that out; this is going to give me x times x2, minus x, minus 6, equals 0.0984

Don't forget to bring this x along as you factor this, because this is going to give us one of the solutions.0995

I have x2 - x - 6 that I need to factor; this is in the form (x + a constant) (x - a constant), because the sign is negative.1002

Factors of 6: 1 and 6, 2 and 3; and I need these to add up to -1, and their signs are going to be opposite.1015

These (2 and 3) are close together, so if I make the larger one negative and the smaller one positive, I am going to get a -1.1026

Therefore, 3 is negative, and 2 is positive.1037

Now, according to the zero product property, if any of these terms (x, x + 2, or x - 3) is 0, then this product will be 0.1042

And it will equal the right side of the equation.1053

Therefore, x equals 0 (and make sure you don't leave this one out, or you will be missing one of the solutions), x + 2 = 0, or x - 3 = 0.1056

So here, I don't have to do anything further with this.1068

I just have that one of the solutions here is x = 0.1070

Here, I have to subtract 2 from both sides to get x = -2; and here, I need to add 3 to both sides.1075

So, solutions: x = 0, x = -2, and x = 3; these are all solutions to this equation.1083

And I solved this by factoring out the greatest common factor, x, factoring this trinomial,1098

and then using the zero product property to solve for x in each of these expressions' terms.1107

Here, I am asked to solve x4 - 256 = 0.1119

This is actually the difference of two squares; this is in the form a2 - b2.1126

So, it is going to factor out to (a + b) (a - b).1133

And if you look at it and think about it this way, x2, squared, is x4.1137

And if you look at this one, 256, and take the square root of that, it is actually 16.1143

Therefore, this is telling me that a equals x2, and b equals 16.1155

So, I can factor it as follows: a + b (that is x2 + 16), times a - b (or x2 - 16).1161

OK, and looking at what I have, I can't do anything else with x2 + 16.1177

But I recognized again, here: I have the difference of two squares; this time, a equals x, and b equals 4.1182

So, it is going to factor out to (x + 4) (x - 4); and these are set equal to 0.1189

Now, I use the zero product property, which is going to tell me that I could have x2 + 16 = 0, x + 4 = 0, or x - 4 = 0.1200

And let's work with these simpler ones first.1214

Simply subtract 4 from both sides to give me x = -4.1217

Add 4 to both sides: x = 4; I have two of my solutions.1224

Now, looking over here, it is a little bit more complex: subtract 16 from both sides--that gives me x2 = -16.1229

Now, I am going to take the square root of both sides, and you can see that this ends up being ±√-16.1239

And this is a negative number; and since I know that √-1 equals i, I can rewrite this as x = ±i√16.1247

Well, the square root of 16 is 4, so this gives me a complex conjugate pair, plus or minus 4i.1264

So, I have four solutions: x = 4i, x = -4i, x = 4, and x = -4.1276

And let's just think about Descartes' Rule of Signs and show that it predicted the possibilities for the type of roots that I could get.1293

Since I have x4 - 256, if I look at the number of sign changes for this, this is positive to negative (one sign change).1303

This tells me that I am going to have one positive real root, or less than that by an even number;1320

but I can't go there, because then I would be going into negative numbers,1327

and I can't say there are -1 real roots; that wouldn't make sense.1330

So, it is just one positive real root.1333

Now, looking at f(-x): this gives me -x4 - 256.1340

Well, this -1, when you take it to the fourth power, is just going to become positive; so this gives me this.1350

And again, I have one sign change; so this tells me that I have one negative real root.1359

Since the degree here is 4, I have 4 total roots.1370

So, this is going to leave me with one positive real, one negative real, and two complex roots,1376

which is exactly what I see: a positive real, a negative real, and the set of complex conjugates.1387

Determine the possible combination of positive real roots, negative real roots, and complex roots.1395

We will use Descartes' Rule of Signs to determine this, and we will start out by thinking about the total.1402

Since the degree is 3, there are 3 total roots.1410

Now, using the rule of signs, I am going to look for f(x) and the sign changes: this is 2x3 - 3x2 + 4x - 5.1421

Number of sign changes: well, this is positive, and this is negative--that is one.1432

I am going from negative here to positive here; that is two; from positive to negative--that is 3.1440

So, the number of sign changes equals 3.1448

Therefore, the number of positive real roots is 3, or less than this by an even number (3 - 2 is 1).1450

You can't go any lower than that; if I subtracted by 2 again, I would get a negative number.1465

So, I have either 3 or 1 positive real roots.1470

All right, now let's look at the negative scenario for f(-x) to figure out the negative real roots.1477

2 times -x cubed, minus 3 times -x squared, plus 4 times -x, minus 5:1484

OK, this is going to give me f(-x) =...this is going to remain negative, so this is going to give me -2x3.1500

A negative and a negative (squared) is going to give me a positive, so this will become x2.1512

This is negative here, though, so it is -3x2.1519

4 times -x is -4x, minus 5.1524

OK, the number of sign changes: none, none, none: the signs are all negative--the coefficients of f(-x).1529

So, the number of sign changes equals 0; so there are 0 negative real roots.1542

OK, so let's figure out what we have going on here.1551

We have positive real roots, negative real roots, and complex.1554

And remember: these need to total 3.1567

Positive real: I could have 3; negative real: 0; I need them to total 3--this already does, so the complex is going to be 0, since this totals 3.1571

OK, another possibility is that I have one positive real root, 0 negative real roots.1586

1 and 0 is 1; I need it to total 3; so there must be 2 complex real roots.1594

So, the possible combination of positive real roots, negative real roots, and complex roots is 3, 0, 0, or 1, 0, 2.1601

And I know I have three total roots, because the degree is 3.1612

I use Descartes' Rule of Signs to tell me that I had either 3 or 1 positive real roots; I have 0 negative real roots.1615

And then, figuring out what is lacking to get my total of 3, I could fill it in with complex roots.1623

OK, determine the possible combinations of positive real roots, negative real roots, and complex roots.1632

Degree equals 4, so I have four total roots.1639

f(x) = -3x4 - 4x3 + 2x2 + 6x + 7.1649

So, positive real roots: let's look for sign changes between these coefficients.1659

A negative to a negative; a negative to a positive (that is 1); a positive to a positive--no sign change; a positive to a positive again.1667

So, the positive real roots equals 1; and I can't go by less than that,1678

because if I subtracted 2, I would go into negative numbers.1685

So, the number of positive real roots is 1.1688

Now, looking at negative real roots: I am going to take f(-x) to get1691

-3 times -x to the fourth, minus 4 times -x to the third, times 2 times -x squared, plus 6 times -x, plus 7.1700

OK, this gives me f(-x) =...well, -1 to the fourth power--this will become positive; so I have -3x41718

Here, -x cubed...this is going to remain negative as a coefficient...times -4; that is going to become positive, so this is going to give me + 4x3.1733

-x squared is going to give me positive x2 times 2; so that is + 2x2; 6 minus -x is -6x; plus 7.1748

OK, negative real roots is going to be determined by the sign changes of the coefficients of f(-x), according to Descartes' Rule of Signs.1760

A negative to a positive; that is one sign change; that stays positive; a positive to a negative: 2; a negative to a positive: 3.1772

So, here I have 3 or less than that by an even number: 3 - 2 is 1.1784

You can't go any less than that.1796

I have a total of 4 roots; I have one positive real root, and either 3 or 1 negative real roots.1799

So, let's look at the possibilities: positive real roots, negative real roots, and complex.1810

Positive real: 1; then I could have a negative real root totaling 3: 1 + 3 is 4; I only have 4; so complex must be 0.1825

Or I could have 1 positive real root and 1 negative real root; and I am going to total those to get 2.1838

But I should have 4; so in this case, there would be two complex roots--a pair of complex conjugates.1847

OK, so I determined there were four total roots, because the degree here is 4.1855

Using Descartes' Rule of Signs, the positive case, f(x), I found that there is one positive real root.1861

f(-x) gave me three sign changes, so there are either three or one negative real roots.1869

And then, to get a total of four, I had zero complex roots in this case, and two in this case.1875

Thanks for visiting Educator.com, and I will see you soon!1884

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