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INSTRUCTORS Carleen Eaton Grant Fraser Eric Smith
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Eric Smith

Eric Smith

Polynomial Inequalities

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Table of Contents

I. Properties of Real Numbers
Basic Types of Numbers

30m 41s

Intro
0:00
Objectives
0:07
Basic Types of Numbers
0:36
Natural Numbers
1:02
Whole Numbers
1:29
Integers
2:04
Rational Numbers
2:38
Irrational Numbers
5:06
Imaginary Numbers
6:48
Basic Types of Numbers Cont.
8:09
The Big Picture
8:10
Real vs. Imaginary Numbers
8:30
Rational vs. Irrational Numbers
8:48
Basic Types of Numbers Cont.
10:55
Number Line
11:06
Absolute Value
11:44
Inequalities
12:39
Example 1
13:16
Example 2
17:30
Example 3
21:56
Example 4
24:27
Example 5
27:48
Operations on Numbers

19m 26s

Intro
0:00
Objectives
0:06
Operations on Numbers
0:25
Addition
0:53
Subtraction
1:33
Multiplication & Division
2:19
Exponents
3:24
Bases
4:04
Square Roots
4:59
Principle Square Roots
5:09
Perfect Squares
6:32
Simplifying and Combining Roots
6:52
Example 1
8:16
Example 2
12:30
Example 3
14:02
Example 4
16:27
Order of Operations

12m 6s

Intro
0:00
Objectives
0:06
The Order of Operations
0:25
Work Inside Parentheses
0:42
Simplify Exponents
0:52
Multiplication & Division from Left to Right
0:57
Addition & Subtraction from Left to Right
1:11
Remember PEMDAS
1:21
The Order of Operations Cont.
2:27
Example
2:43
Example 1
3:55
Example 2
5:36
Example 3
7:35
Example 4
8:56
Properties of Real Numbers

18m 52s

Intro
0:00
Objectives
0:07
The Properties of Real Numbers
0:23
Commutative Property of Addition and Multiplication
0:44
Associative Property of Addition and Multiplication
1:50
Distributive Property of Multiplication Over Addition
3:20
Division Property of Zero
4:46
Division Property of One
5:23
Multiplication Property of Zero
5:56
Multiplication Property of One
6:17
Addition Property of Zero
6:29
Why Are These Properties Important?
6:53
Example 1
9:16
Example 2
13:04
Example 3
14:30
Example 4
16:57
II. Linear Equations
The Vocabulary of Linear Equations

12m 22s

Intro
0:00
Objectives
0:09
The Vocabulary of Linear Equations
0:44
Variables
0:52
Terms
1:09
Coefficients
1:40
Like Terms
2:18
Examples of Like Terms
2:37
Expressions
4:01
Equations
4:26
Linear Equations
5:04
Solutions
5:55
Example 1
6:16
Example 2
7:16
Example 3
8:45
Example 4
10:20
Solving Linear Equations in One Variable

28m 52s

Intro
0:00
Objectives
0:08
Solving Linear Equations in One Variable
0:34
Conditional Cases
0:51
Identity Cases
1:09
Contradiction Cases
1:30
Solving Linear Equations in One Variable Cont.
2:00
Addition Property of Equality
2:10
Multiplication Property of Equality
2:43
Steps to Solve Linear Equations
3:14
Example 1
4:22
Example 2
8:21
Example 3
12:32
Example 4
14:19
Example 5
17:25
Example 6
22:17
Solving Formulas

12m 2s

Intro
0:00
Objectives
0:06
Solving Formulas
0:18
Formulas
0:26
Use the Same Properties as Solving Linear Equations
1:36
Addition Property of Equality
1:55
Multiplication Property of Equality
1:58
Steps to Solve Formulas
2:43
Example 1
3:56
Example 2
6:09
Example 3
8:39
Applications of Linear Equations

28m 41s

Intro
0:00
Objectives
0:10
Applications of Linear Equations
0:43
The Six-Step Method to Solving Word Problems
0:55
Common Terms
3:12
Example 1
5:03
Example 2
9:40
Example 3
13:48
Example 4
17:58
Example 5
23:28
Applications of Linear Equations, Motion & Mixtures

24m 26s

Intro
0:00
Objectives
0:21
Motion and Mixtures
0:46
Motion Problems: Distance, Rate, and Time
1:06
Mixture Problems: Amount, Percent, and Total
1:27
The Table Method
1:58
The Beaker Method
3:38
Example 1
5:05
Example 2
9:44
Example 3
14:20
Example 4
19:13
III. Graphing
Rectangular Coordinate System

22m 55s

Intro
0:00
Objectives
0:11
The Rectangular Coordinate System
0:39
The Cartesian Coordinate System
0:40
X-Axis
0:54
Y-Axis
1:04
Origin
1:11
Quadrants
1:26
Ordered Pairs
2:10
Example 1
2:55
The Rectangular Coordinate System Cont.
6:09
X-Intercept
6:45
Y-Intercept
6:55
Relation of X-Values and Y-Values
7:30
Example 2
11:03
Example 3
12:13
Example 4
14:10
Example 5
18:38
Slope & Graphing

27m 58s

Intro
0:00
Objectives
0:11
Slope and Graphing
0:48
Standard Form
1:14
Example 1
2:24
Slope and Graphing Cont.
4:58
Slope, m
5:07
Slope is Rise over Run
6:11
Don't Mix Up the Coordinates
8:20
Example 2
9:39
Slope and Graphing Cont.
14:26
Slope-Intercept Form
14:34
Example 3
16:55
Example 4
18:00
Slope and Graphing Cont.
19:00
Rewriting an Equation in Slope-Intercept Form
19:39
Rewriting an Equation in Standard Form
20:09
Slopes of Vertical & Horizontal Lines
20:56
Example 5
22:49
Example 6
24:09
Example 7
25:59
Example 8
26:57
Linear Equations in Two Variables

20m 36s

Intro
0:00
Objectives
0:13
Linear Equations in Two Variables
0:36
Point-Slope Form
1:07
Substitute in the Point and the Slope
2:21
Parallel Lines: Two Lines with the Same Slope
4:05
Perpendicular Lines: Slopes are Negative Reciprocals of Each Other
4:39
Perpendicular Lines: Product of Slopes is -1
5:24
Example 1
6:02
Example 2
7:50
Example 3
10:49
Example 4
13:26
Example 5
15:30
Example 6
17:43
IV. Functions
Introduction to Functions

21m 24s

Intro
0:00
Objectives
0:07
Introduction to Functions
0:58
Relations
1:03
Functions
1:37
Independent Variables
2:00
Dependent Variables
2:11
Function Notation
2:21
Function
3:43
Input and Output
3:53
Introduction to Functions Cont.
4:45
Domain
4:46
Range
4:55
Functions Represented by a Diagram
6:41
Natural Domain
9:11
Evaluating Functions
12:02
Example 1
13:13
Example 2
15:03
Example 3
16:18
Example 4
19:54
Graphing Functions

16m 12s

Intro
0:00
Objectives
0:09
Graphing Functions
0:54
Using Slope-Intercept Form
1:56
Vertical Line Test
2:58
Determining the Domain
4:20
Determining the Range
5:43
Example 1
6:06
Example 2
7:18
Example 3
8:31
Example 4
11:04
V. Systems of Linear Equations
Systems of Linear Equations

25m 54s

Intro
0:00
Objectives
0:13
Systems of Linear Equations
0:46
System of Equations
0:51
System of Linear Equations
1:15
Solutions
1:35
Points as Solutions
1:53
Finding Solutions Graphically
5:13
Example 1
6:37
Example 2
12:07
Systems of Linear Equations Cont.
17:01
One Solution, No Solution, or Infinite Solutions
17:10
Example 3
18:31
Example 4
22:37
Solving a System Using Substitution

20m 1s

Intro
0:00
Objectives
0:09
Solving a System Using Substitution
0:32
Substitution Method
1:24
Substitution Example
2:35
One Solution, No Solution, or Infinite Solutions
7:50
Example 1
9:45
Example 2
12:48
Example 3
15:01
Example 4
17:30
Solving a System Using Elimination

19m 40s

Intro
0:00
Objectives
0:09
Solving a System Using Elimination
0:27
Elimination Method
0:42
Elimination Example
2:01
One Solution, No Solution, or Infinite Solutions
7:05
Example 1
8:53
Example 2
11:46
Example 3
15:37
Example 4
17:45
Applications of Systems of Equations

24m 34s

Intro
0:00
Objectives
0:12
Applications of Systems of Equations
0:30
Word Problems
1:31
Example 1
2:17
Example 2
7:55
Example 3
13:07
Example 4
17:15
VI. Inequalities
Solving Linear Inequalities in One Variable

17m 13s

Intro
0:00
Objectives
0:08
Solving Linear Inequalities in One Variable
0:37
Inequality Expressions
0:46
Linear Inequality Solution Notations
3:40
Inequalities
3:51
Interval Notation
4:04
Number Lines
4:43
Set Builder Notation
5:24
Use Same Techniques as Solving Equations
6:59
'Flip' the Sign when Multiplying or Dividing by a Negative Number
7:12
'Flip' Example
7:50
Example 1
8:54
Example 2
11:40
Example 3
14:01
Compound Inequalities

16m 13s

Intro
0:00
Objectives
0:07
Compound Inequalities
0:37
'And' vs. 'Or'
0:44
'And'
3:24
'Or'
3:35
'And' Symbol, or Intersection
3:51
'Or' Symbol, or Union
4:13
Inequalities
4:41
Example 1
6:22
Example 2
9:30
Example 3
11:27
Example 4
13:49
Solving Equations with Absolute Values

14m 12s

Intro
0:00
Objectives
0:08
Solve Equations with Absolute Values
0:18
Solve Equations with Absolute Values Cont.
1:11
Steps to Solving Equations with Absolute Values
2:21
Example 1
3:23
Example 2
6:34
Example 3
10:12
Inequalities with Absolute Values

17m 7s

Intro
0:00
Objectives
0:07
Inequalities with Absolute Values
0:23
Recall…
2:08
Example 1
3:39
Example 2
6:06
Example 3
8:14
Example 4
10:29
Example 5
13:29
Graphing Inequalities in Two Variables

15m 33s

Intro
0:00
Objectives
0:07
Graphing Inequalities in Two Variables
0:32
Split Graph into Two Regions
1:53
Graphing Inequalities
5:44
Test Points
6:20
Example 1
7:11
Example 2
10:17
Example 3
13:06
Systems of Inequalities

21m 13s

Intro
0:00
Objectives
0:08
Systems of Inequalities
0:24
Test Points
1:10
Steps to Solve Systems of Inequalities
1:25
Example 1
2:23
Example 2
7:28
Example 3
12:51
VII. Polynomials
Integer Exponents

44m 51s

Intro
0:00
Objectives
0:09
Integer Exponents
0:42
Exponents 'Package' Multiplication
1:25
Example 1
2:00
Example 2
3:13
Integer Exponents Cont.
4:50
Product Rule for Exponents
4:51
Example 3
7:16
Example 4
10:15
Integer Exponents Cont.
13:13
Power Rule for Exponents
13:14
Power Rule with Multiplication and Division
15:33
Example 5
16:18
Integer Exponents Cont.
20:04
Example 6
20:41
Integer Exponents Cont.
25:52
Zero Exponent Rule
25:53
Quotient Rule
28:24
Negative Exponents
30:14
Negative Exponent Rule
32:27
Example 7
34:05
Example 8
36:15
Example 9
39:33
Example 10
43:16
Adding & Subtracting Polynomials

18m 33s

Intro
0:00
Objectives
0:07
Adding and Subtracting Polynomials
0:25
Terms
0:33
Coefficients
0:51
Leading Coefficients
1:13
Like Terms
1:29
Polynomials
2:21
Monomials, Binomials, Trinomials, and Polynomials
5:41
Degrees
7:00
Evaluating Polynomials
8:12
Adding and Subtracting Polynomials Cont.
9:25
Example 1
11:48
Example 2
13:00
Example 3
14:41
Example 4
16:15
Multiplying Polynomials

25m 7s

Intro
0:00
Objectives
0:06
Multiplying Polynomials
0:41
Distributive Property
1:00
Example 1
2:49
Multiplying Polynomials Cont.
8:22
Organize Terms with a Table
8:23
Example 2
13:40
Multiplying Polynomials Cont.
16:33
Multiplying Binomials with FOIL
16:48
Example 3
18:49
Example 4
20:04
Example 5
21:42
Dividing Polynomials

44m 56s

Intro
0:00
Objectives
0:07
Dividing Polynomials
0:29
Dividing Polynomials by Monomials
2:10
Dividing Polynomials by Polynomials
2:59
Dividing Numbers
4:09
Dividing Polynomials Example
8:39
Example 1
12:35
Example 2
14:40
Example 3
16:45
Example 4
21:13
Example 5
24:33
Example 6
29:02
Dividing Polynomials with Synthetic Division Method
33:36
Example 7
38:43
Example 8
42:24
VIII. Factoring Polynomials
Greatest Common Factor & Factor by Grouping

28m 27s

Intro
0:00
Objectives
0:09
Greatest Common Factor
0:31
Factoring
0:40
Greatest Common Factor (GCF)
1:48
GCF for Polynomials
3:28
Factoring Polynomials
6:45
Prime
8:21
Example 1
9:14
Factor by Grouping
14:30
Steps to Factor by Grouping
17:03
Example 2
17:43
Example 3
19:20
Example 4
20:41
Example 5
22:29
Example 6
26:11
Factoring Trinomials

21m 44s

Intro
0:00
Objectives
0:06
Factoring Trinomials
0:25
Recall FOIL
0:26
Factor a Trinomial by Reversing FOIL
1:52
Tips when Using Reverse FOIL
5:31
Example 1
7:04
Example 2
9:09
Example 3
11:15
Example 4
13:41
Factoring Trinomials Cont.
15:50
Example 5
18:42
Factoring Trinomials Using the AC Method

30m 9s

Intro
0:00
Objectives
0:08
Factoring Trinomials Using the AC Method
0:27
Factoring when Leading Term has Coefficient Other Than 1
1:07
Reversing FOIL
1:18
Example 1
1:46
Example 2
4:28
Factoring Trinomials Using the AC Method Cont.
7:45
The AC Method
8:03
Steps to Using the AC Method
8:19
Tips on Using the AC Method
9:29
Example 3
10:45
Example 4
16:50
Example 5
21:08
Example 6
24:58
Special Factoring Techniques

30m 14s

Intro
0:00
Objectives
0:07
Special Factoring Techniques
0:26
Difference of Squares
1:46
Perfect Square Trinomials
2:38
No Sum of Squares
3:32
Special Factoring Techniques Cont.
4:03
Difference of Squares Example
4:04
Perfect Square Trinomials Example
5:29
Example 1
7:31
Example 2
9:59
Example 3
11:47
Example 4
15:09
Special Factoring Techniques Cont.
19:07
Sum of Cubes and Difference of Cubes
19:08
Example 5
23:13
Example 6
26:12
IX. Quadratic Equations
Solving Quadratic Equations by Factoring

23m 38s

Intro
0:00
Objectives
0:08
Solving Quadratic Equations by Factoring
0:19
Quadratic Equations
0:20
Zero Factor Property
1:39
Zero Factor Property Example
2:34
Example 1
4:00
Solving Quadratic Equations by Factoring Cont.
5:54
Example 2
7:28
Example 3
11:09
Example 4
14:22
Solving Quadratic Equations by Factoring Cont.
18:17
Higher Degree Polynomial Equations
18:18
Example 5
20:22
Solving Quadratic Equations

29m 27s

Intro
0:00
Objectives
0:12
Solving Quadratic Equations
0:29
Linear Factors
0:38
Not All Quadratics Factor Easily
1:22
Principle of Square Roots
3:36
Completing the Square
4:50
Steps for Using Completing the Square
5:15
Completing the Square Works on All Quadratic Equations
6:41
The Quadratic Formula
7:28
Discriminants
8:25
Solving Quadratic Equations - Summary
10:11
Example 1
11:54
Example 2
13:03
Example 3
16:30
Example 4
21:29
Example 5
25:07
Equations in Quadratic Form

16m 47s

Intro
0:00
Objectives
0:08
Equations in Quadratic Form
0:24
Using a Substitution
0:53
U-Substitution
1:26
Example 1
2:07
Example 2
5:36
Example 3
8:31
Example 4
11:14
Quadratic Formulas & Applications

29m 4s

Intro
0:00
Objectives
0:09
Quadratic Formulas and Applications
0:35
Squared Variable
0:40
Principle of Square Roots
0:51
Example 1
1:09
Example 2
2:04
Quadratic Formulas and Applications Cont.
3:34
Example 3
4:42
Example 4
13:33
Example 5
20:50
Graphs of Quadratics

26m 53s

Intro
0:00
Objectives
0:06
Graphs of Quadratics
0:39
Axis of Symmetry
1:46
Vertex
2:12
Transformations
2:57
Graphing in Quadratic Standard Form
3:23
Example 1
5:06
Example 2
6:02
Example 3
9:07
Graphs of Quadratics Cont.
11:26
Completing the Square
12:02
Vertex Shortcut
12:16
Example 4
13:49
Example 5
17:25
Example 6
20:07
Example 7
23:43
Polynomial Inequalities

21m 42s

Intro
0:00
Objectives
0:07
Polynomial Inequalities
0:30
Solving Polynomial Inequalities
1:20
Example 1
2:45
Polynomial Inequalities Cont.
5:12
Larger Polynomials
5:13
Positive or Negative Intervals
7:16
Example 2
9:01
Example 3
13:53
X. Rational Equations
Multiply & Divide Rational Expressions

26m 41s

Intro
0:00
Objectives
0:09
Multiply and Divide Rational Expressions
0:44
Rational Numbers
0:55
Dividing by Zero
1:45
Canceling Extra Factors
2:43
Negative Signs in Fractions
4:52
Multiplying Fractions
6:26
Dividing Fractions
7:17
Example 1
8:04
Example 2
14:01
Example 3
16:23
Example 4
18:56
Example 5
22:43
Adding & Subtracting Rational Expressions

20m 24s

Intro
0:00
Objectives
0:07
Adding and Subtracting Rational Expressions
0:41
Common Denominators
0:52
Common Denominator Examples
1:14
Steps to Adding and Subtracting Rational Expressions
2:39
Example 1
3:34
Example 2
5:27
Adding and Subtracting Rational Expressions Cont.
6:57
Least Common Denominators
6:58
Transitioning from Fractions to Rational Expressions
9:08
Identifying Least Common Denominators for Rational Expressions
9:56
Subtracting vs. Adding
10:41
Example 3
11:19
Example 4
12:36
Example 5
15:08
Example 6
16:46
Complex Fractions

18m 23s

Intro
0:00
Objectives
0:09
Complex Fractions
0:37
Dividing to Simplify Complex Fractions
1:10
Example 1
2:03
Example 2
3:58
Complex Fractions Cont.
9:15
Using the Least Common Denominator to Simplify Complex Fractions
9:16
Both Methods Lead to the Same Answer
10:07
Example 3
10:42
Example 4
14:28
Solving Rational Equations

16m 24s

Intro
0:00
Objectives
0:07
Solving Rational Equations
0:23
Isolate the Specified Variable
1:23
Example 1
1:58
Example 2
5:00
Example 3
8:23
Example 4
13:25
Rational Inequalities

18m 54s

Intro
0:00
Objectives
0:06
Rational Inequalities
0:18
Testing Intervals for Rational Inequalities
0:38
Steps to Solving Rational Inequalities
1:05
Tips to Solving Rational Inequalities
2:27
Example 1
3:33
Example 2
12:21
Applications of Rational Expressions

20m 20s

Intro
0:00
Objectives
0:07
Applications of Rational Expressions
0:27
Work Problems
1:05
Example 1
2:58
Example 2
6:45
Example 3
13:17
Example 4
16:37
Variation & Proportion

27m 4s

Intro
0:00
Objectives
0:10
Variation and Proportion
0:34
Variation
0:35
Inverse Variation
1:01
Direct Variation
1:10
Setting Up Proportions
1:31
Example 1
2:27
Example 2
5:36
Variation and Proportion Cont.
8:29
Inverse Variation
8:30
Example 3
9:20
Variation and Proportion Cont.
12:41
Constant of Proportionality
12:42
Example 4
13:59
Variation and Proportion Cont.
16:17
Varies Directly as the nth Power
16:30
Varies Inversely as the nth Power
16:53
Varies Jointly
17:09
Combining Variation Models
17:36
Example 5
19:09
Example 6
22:10
XI. Radical Equations
Rational Exponents

14m 32s

Intro
0:00
Objectives
0:07
Rational Exponents
0:32
Power on Top, Root on Bottom
1:05
Example 1
1:37
Rational Exponents Cont.
4:04
Using Rules from Exponents for Radicals as Exponents
4:05
Combining Terms Under a Single Root
4:50
Example 2
5:21
Example 3
7:39
Example 4
11:23
Example 5
13:14
Simplify Rational Exponents

15m 12s

Intro
0:00
Objectives
0:07
Simplify Rational Exponents
0:25
Product Rule for Radicals
0:26
Product Rule to Simplify Square Roots
1:11
Quotient Rule for Radicals
1:42
Applications of Product and Quotient Rules
2:17
Higher Roots
2:48
Example 1
3:39
Example 2
6:35
Example 3
8:41
Example 4
11:09
Adding & Subtracting Radicals

17m 22s

Intro
0:00
Objectives
0:07
Adding and Subtracting Radicals
0:33
Like Terms
1:29
Bases and Exponents May be Different
2:02
Bases and Powers Must be Same when Adding and Subtracting
2:42
Add Radicals' Coefficients
3:55
Example 1
4:47
Example 2
6:00
Adding and Subtracting Radicals Cont.
7:10
Simplify the Bases to Look the Same
7:25
Example 3
8:23
Example 4
11:45
Example 5
15:10
Multiply & Divide Radicals

19m 24s

Intro
0:00
Objectives
0:08
Multiply and Divide Radicals
0:25
Rules for Working With Radicals
0:26
Using FOIL for Radicals
1:11
Don’t Distribute Powers
2:54
Dividing Radical Expressions
4:25
Rationalizing Denominators
6:40
Example 1
7:22
Example 2
8:32
Multiply and Divide Radicals Cont.
9:23
Rationalizing Denominators with Higher Roots
9:25
Example 3
10:51
Example 4
11:53
Multiply and Divide Radicals Cont.
13:13
Rationalizing Denominators with Conjugates
13:14
Example 5
15:52
Example 6
17:25
Solving Radical Equations

15m 5s

Intro
0:00
Objectives
0:07
Solving Radical Equations
0:17
Radical Equations
0:18
Isolate the Roots and Raise to Power
0:34
Example 1
1:13
Example 2
3:09
Solving Radical Equations Cont.
7:04
Solving Radical Equations with More than One Radical
7:05
Example 3
7:54
Example 4
13:07
Complex Numbers

29m 16s

Intro
0:00
Objectives
0:06
Complex Numbers
1:05
Imaginary Numbers
1:08
Complex Numbers
2:27
Real Parts
2:48
Imaginary Parts
2:51
Commutative, Associative, and Distributive Properties
3:35
Adding and Subtracting Complex Numbers
4:04
Multiplying Complex Numbers
6:16
Dividing Complex Numbers
8:59
Complex Conjugate
9:07
Simplifying Powers of i
14:34
Shortcut for Simplifying Powers of i
18:33
Example 1
21:14
Example 2
22:15
Example 3
23:38
Example 4
26:33
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Lecture Comments (3)

0 answers

Post by Sage Stark on August 12, 2017

Also, in example three, why did you change -5/4 to -2, it didn't here, but it could potentially alter the solution if the situation was right.

1 answer

Last reply by: Professor Eric Smith
Sun Dec 3, 2017 5:52 PM

Post by Sage Stark on August 12, 2017

In example three, why is the five positive and the three is negative? I thought it would be the other way around.

Polynomial Inequalities

  • Remember that the solution to an inequality often involves a range of values.
  • To solve an inequality involving polynomials we
    • Set the inequality with zero on one side
    • Solve the related equation, with the polynomial equal to zero
    • Divide the x-axis into intervals using the solutions of the equation from step 2
    • Use test values from each interval to see if it satisfies the inequality
    • Check the endpoints of each interval to see if they need to be included
  • Do not attempt to split up the inequality over the factors. Even though it looks like we can use the principle of zero products we can’t. This is because we have an inequality and the principle of zero products only applies to equations.

Polynomial Inequalities

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
  • Objectives 0:07
  • Polynomial Inequalities 0:30
    • Solving Polynomial Inequalities
  • Example 1 2:45
  • Polynomial Inequalities Cont. 5:12
    • Larger Polynomials
    • Positive or Negative Intervals
  • Example 2 9:01
  • Example 3 13:53

Transcription: Polynomial Inequalities

Welcome back to www.educator.com.0000

In this lesson we are going to take a look at some polynomial inequalities.0002

When it comes to polynomial inequalities, I’m going to show you two techniques that you can use to handle these.0008

One, you can take a look and figure out their graph and figure out where they are greater than or equal to 00015

or you can use a table to tracked down their sign.0020

Both of these are important and since we have developed lots of graphing techniques they are both very handy.0023

A polynomial inequality is simply an inequality involving a polynomial of some sort.0033

Down below I have many different examples of what I'm talking about.0040

Here is a nice -2x4 - x2 - 2 > 4 that will be an example of something we are trying to solve.0044

With this one, 8x3 + 2x2 < 6x + 5 is another good type of inequality that we could possibly find the solution to.0054

Depending on the type of polynomial present, sometimes you can call these linear inequalities, quadratic inequalities or even cubic inequalities.0064

I will not get too technical usually we will just call these like polynomial inequalities.0073

To solve one of these inequalities, it is best to get it in relation to 0.0081

It means get everything over onto one side of your inequality symbol.0088

The reason why this is so important for the entire solving process is we are looking for those ranges of values that make the inequality true.0093

When it is all in relation to 0 then we just have to know where it is going to be positive or negative0101

rather than trying to search for when it is greater, less than a some other number like 2 or 3.0106

That is much more difficult to do.0111

If you have a graph of the polynomial, this makes it especially easy because then you are essentially looking at whether it is above or below the x axis.0114

Let us take a linear equality like this.0128

We might know where it crosses the x axis everywhere where it is above the x axis we know it is positive.0132

Everywhere below the x axis we know the value is negative.0141

If I had the equation for this and maybe I wanted to know where it was greater than 00147

then I can simply take all of the positive values where it is greater than the x axis.0157

Let us try a quick example to see this in action.0167

I want to use the graph of x2 - 3x -4 to see where it is greater than 0.0171

It is already said in relation to 0 for us, we do not have to work there.0178

I just need a good rough sketch of what this looks like.0180

I can figure out where this thing is equal to 0 by using some of our factoring techniques.0186

Our first terms better be x and then - 4 + 1 will do fine.0193

I know it crosses the x axis at 4 and -1.0199

Since we have also done a lot with graphing quadratic like this one, we know which direction it is facing.0206

The (a) value is positive, so it is facing up.0212

Let us see what else do we know about this?0218

It looks like it has a y intercept of -4.0221

Let us put that in there.0226

A rough sketch of this might look something like that.0231

You will note that I did not go through all of the work to find the vertex because I’m not interested where it reaches its maximum or minimum value.0238

I'm interested in where is it above or below the x axis.0245

For this particular polynomial we want to know where it is greater than 0 because that is a way of saying where does it take on positive values.0258

We can see that it does this after the value of 4 and does it before the value of -1.0269

In those intervals, if you are using saying those x values, then the equation will be positive.0277

We will take those intervals as our solution, from negative infinity up to -1, and from 4 up to infinity.0289

That grabs both of those intervals.0301

We will use out little union symbol in between the show that it could be in either one of those.0304

That was a quadratic example but potentially you could use these for some much larger polynomials.0315

The key is creating an accurate enough graphs so we can see whether it is above or below the x axis.0321

You want it in relation to 0 to see what it is doing.0328

I’m just going to give a rough sketch of a polynomial, so you can see how this process might work.0331

Let us call this my polynomial and I want to know where this one is less than or equal to 0.0341

Everywhere it is above the x axis would be positive and everywhere below the x axis it takes on negative values.0351

Look at these portions right here, here is below the x axis.0360

It has another little part right there and this last term.0366

We can highlight the intervals where you take on these negative values.0372

Here, here and here.0377

Okay, now we just have to describe those intervals using our interval notation.0383

It is below the x axis from negative infinity all the way up to 1, 2, 3, 4, 5, -5 then it goes from -1 up to 3.0388

I got the 3, 4, 5 from 5 up to infinity.0405

We will use our little union symbol to connect all of those intervals.0413

One quick note, we are including these end points because it takes on the value of 0 at these points.0417

We want them included.0427

Having a graph is handy.0437

It gives us a nice visual way to see what is going on whether it is greater than 0 or less than 0,0439

but unfortunately we do not always have a graph to work with.0444

I’m thinking of some much more complicated polynomials.0447

In situations like this, we are going to use a table to keep track of the sign of its factors.0450

That way we will still be able to figure out whether it is above or below the x axis without ever seeing the graph.0455

Here is how this process works.0461

We will first get the inequality in relation to 0, so we will get everything over onto one side.0464

Then we will go ahead and solve the related equation.0471

Instead of looking at the inequality we will throw in an equal sign in there and then solve it and figure out where it is equal to 0.0474

The reason why we are doing this is we want know where it could possibly change sign.0481

Where does it hit the x axis?0485

That will allow us to divide up the x axis into a lot of different intervals using those points.0488

We will use our values around those points to see whether it is positive or negative in those factors.0495

Once we have a lot of information about where is positive or negative0504

we will determine which of those intervals actually satisfy the inequality.0507

We will check the end points of those intervals to see if we need to include them or not.0513

If we see something like or equals, we will usually include the values of our end points using brackets.0520

If it is a nice strict inequality, then we will usually use parentheses to say that those end points are not included.0532

Let us borrow a problem that we did earlier for quadratic and see how the same process works out now using a table.0542

The first thing we want is we want it in relation to 0, which fortunately it is already is.0552

We do not have to worry about that part.0558

I’m going to look at the corresponding equation.0561

Where could this thing equal 0?0563

I think we factored this one earlier it is going to come in handy once more.0568

I have x - 4 and x + 1.0572

There are two values this could equal 0, at 4 and -1.0578

Here is where our table comes into play.0585

I’m going to make a number line and I’m going to mark out the location of -1 and the location of 4.0587

You can see that t does split up our number line into these different intervals.0596

I need to figure out what is the polynomial doing on these different intervals.0602

I’m going to write down the factors of the polynomial and just experiment0608

to see whether they are ending up being positive or negative on these intervals.0615

That is what our test points are for.0621

Let us give this a try.0623

I’m going to choose something on my number line that is less than -1.0625

Think of something like -2, something on that side.0630

We are going to test it into these factors over here.0634

If I was to plug in -2 in for x and then I subtract a 4, will that give me a positive value or negative value?0639

-2 -4 will be -6 and I know on this interval that factor is a negative.0648

What if I took -2 and I put in the second factor, and I added 1 to it.0657

-2 + 1 I can see that that give me -1.0662

Of course, the most important part is that factor would be negative on that interval.0667

I’m keeping track of the individual parts of the polynomial seeing whether positive or negative.0672

Now that we have a good idea what is going on that interval, let us pick something on the next interval.0679

Something between -1 and 4.0684

Since I have worked out good, let us test out 0 and see what it does.0688

0 - 4 will that be positive or negative?0693

That will give us -4.0697

Let us plug in 0 into x + 1 and that will give us 1.0701

I know that it is positive on that interval.0706

Now something larger than 4, let us test out something over there.0711

Let us test out something like 5.0715

We will put that into each of our factors, 5 - 4 will give us 1 and 5 + 1 will give us 6.0718

We have lots of information about what each of these individual factors are doing on these intervals.0733

Of course, if we look back at the original problem that we are looking at, these factors are multiplied together.0739

Let us multiply these individual pieces together and least their signs and see what is happening to the overall polynomial.0746

If I were to take a negative × a negative, I would get a positive value.0760

If I were to take a negative × a positive and I will get a negative value.0766

If I multiplied two positives together, I will get a positive value.0772

This is the sign of the overall polynomial when I put those pieces together.0777

What are my interests in, positive or negative values?0783

I'm looking for when the polynomial is greater than 0, so I want those positive values.0788

We will write down the value of the intervals where it was positive.0802

The things greater than 4 and things less than -1.0807

Note how I'm not including the end points here because we are dealing with a strict inequality.0821

This would represent the solution to our polynomial inequality.0827

Of course we might be doing with some more complicated polynomials and that is why we are using these techniques of a table.0834

I have 4x3 - 7x is less than or equal to 15x.0841

I have picked up a lot of the techniques for graphing this in a nice easy way other than just making a giant table of values.0846

Let us use our table here, we would be able to figure out what the interval or range of solutions is for this one.0853

I'm going to get everything over onto one side first.0861

It looks like I'm looking where this is less than or equal to 0 or below that x axis.0872

We want to solve where this thing is equal to 0.0879

We will have to borrow some of our factoring techniques to help out.0886

Everything has an x in common.0891

Now we got all of those taken out.0904

We can go ahead and factor the remaining quadratic.0908

What values could I use in here?0918

I have a 4x, 1x, 5, -3.0921

First terms will give us 4x2, outside would be -12, inside would be 5 and our last terms -15.0931

This checks out.0938

x = 0, 4x + 5= 0 and x – 3 = 0.0941

We would end up solving each of these separately.0950

This middle one, let us go ahead and move the 5 over and then divide by 4.0954

For the third one we will simply add 3 to both sides.0961

I have these 3 points where our polynomial could end up changing sign because that is where it is equal to 0.0968

Let us go ahead and draw our number line and put these values on there.0976

That way we can see how it divides up our number line.0980

The smallest thing we have here is -5/4.0985

The next largest would be 0.0990

The largest value we have is 3.0993

Always put these from smallest to largest.0997

Let us also keep track of the pieces and our polynomials.1011

We have x, 4x + 5 and we have x – 3.1017

We are going to use our test value into these individual parts and put them together in the end.1025

Let us start off with our first test value.1032

We need something that is less than -5/4.1034

I know -5/4 is close to -1 I’m going to choose something a little bit farther.1039

It looks like I’m going to choose -2 and put it into all of our factors.1046

If I put in -2 in 4x I will get -2.1050

If I put in -2 in for 4 then 4x + 5 that will give me -x + 5 which would still be negative.1056

Last that I could take that -2 and put it down into x -3, it will give me -5.1064

Negative again.1071

Let us choose something between -5/4 and 0.1076

Here is where that -1 will come into play.1080

Putting a negative 1 in for x will give us a negative value.1084

We need in 4 for 4x + 5 will give us 1.1089

Putting in for the last one, a -1 -3 = -4 so minus down there.1095

That takes care of that.1103

Let us see what happens when we put in 2 for all of these spots.1110

2 and 4x that will be positive.1115

Putting in 2 for 4x + 5 that will be 8 + 5 =13.1119

If I put in 2 for the last one, that will be 2 -3 = -1.1126

Negative.1129

One more last thing, things that are greater than 3.1139

I’m going to test that out with 4.1143

Let us see if we can put that in all the spots.1147

I’m putting in for x and I will get 4 × 4 + 5 = 21 and 4 - 3 =1.1150

There would be that one.1160

That represents all of the individual components in the polynomial and what their sign will be on those intervals.1164

We want to imagine all of these being put together since all of these factors are being multiplied.1170

Let us do this one column at a time.1178

Negative × negative × another negative, that value would be negative.1181

We have a negative × a positive and that will be negative, times another negative and that would be positive.1190

Now we have a positive × positive × negative, that is negative.1201

Our last column are all positives being multiplied together so positive.1207

I have a good sense of what the overall polynomial is doing for these different intervals.1213

I can see when it is negative, positive, negative and positive.1220

This problem right here, we want to know whether it is less than or equal to 0.1224

We are looking for the negative intervals.1231

You have this interval down here and we have this interval right here.1236

We can go back and highlight those on a number line.1241

I'm looking at the values all the way up to -5/4 and I’m looking at the values between 0 and 3.1246

Let us write down those intervals.1253

negative infinity up to -5/4 and from 0 to 3.1255

Note that I'm definitely including my end points here because this says or equal to 0.1266

I'm also interested in where my polynomial is equal to 0.1274

This would be our intervals solution.1280

No matter what method you use, it is often very important that you know where your polynomial is equal to 0.1284

Use many of our factoring techniques to help you out for that.1290

Then you can go ahead and look at the graph or even one of these tables to organize your information.1294

That will make your life much easier.1298

Thank you for watching www.educator.com.1300

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