INSTRUCTORS Carleen Eaton Grant Fraser

Home » Mathematics » Algebra 2 » Remainder and Factor Theorems

Dr. Carleen Eaton

Dr. Carleen Eaton

Remainder and Factor Theorems

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Table of Contents

Section 1: Equations and Inequalities
Expressions and Formulas

22m 23s

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

20m 15s

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

19m 10s

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

17m 31s

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

17m 14s

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

25m

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

32m 5s

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

14m 46s

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

23m 7s

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

23m 5s

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

31m 5s

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

21m 42s

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

17m 13s

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

23m 53s

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

27m 12s

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

28m 53s

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

11m 34s

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

21m 36s

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

29m 36s

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

33m 13s

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

28m 25s

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

22m 25s

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

22m 32s

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

31m 48s

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

27m 3s

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

19m 53s

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

35m 45s

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

27m 11s

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

22m 48s

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

30m 7s

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

27m 5s

Intro
0:00
Graphing Quadratic Inequalities
0:11
Test Point
0:18
Example: Quadratic Inequality
0:29
Solving Quadratic Inequalities
3:57
Example: Parameter
4:24
Example 1: Graph Inequality
11:16
Example 2: Solve Inequality
14:27
Example 3: Graph Inequality
19:14
Example 4: Solve Inequality
23:48
Section 6: Polynomial Functions
Properties of Exponents

19m 29s

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

13m 27s

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

31m 11s

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

22m 30s

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

33m 29s

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

21m 10s

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

31m 21s

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

31m 27s

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

31m 16s

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

34m 30s

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

22m 42s

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

30m 4s

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

20m 46s

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

41m 11s

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

30m 45s

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

31m 27s

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

40m 54s

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

55m 4s

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

57m 13s

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

20m 21s

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

55m 14s

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

35m 58s

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

45m 54s

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

28m 43s

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

25m 23s

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

21m 14s

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

24m 30s

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

32m 42s

Intro
0:00
Midpoint Formula
0:15
Example: Midpoint
0:30
Distance Formula
2:30
Example: Distance
2:52
Example 1: Midpoint and Distance
4:58
Example 2: Midpoint and Distance
8:07
Example 3: Median Length
18:51
Example 4: Perimeter and Area
23:36
Parabolas

41m 27s

Intro
0:00
What is a Parabola?
0:20
Definition of a Parabola
0:29
Focus
0:59
Directrix
1:15
Axis of Symmetry
3:08
Vertex
3:33
Minimum or Maximum
3:44
Standard Form
4:59
Horizontal Parabolas
5:08
Vertex Form
5:19
Upward or Downward
5:41
Example: Standard Form
6:06
Graphing Parabolas
8:31
Shifting
8:51
Example: Completing the Square
9:22
Symmetry and Translation
12:18
Example: Graph Parabola
12:40
Latus Rectum
17:13
Length
18:15
Example: Latus Rectum
18:35
Horizontal Parabolas
18:57
Not Functions
20:08
Example: Horizontal Parabola
21:21
Focus and Directrix
24:11
Horizontal
24:48
Example 1: Parabola Standard Form
25:12
Example 2: Graph Parabola
30:00
Example 3: Graph Parabola
33:13
Example 4: Parabola Equation
37:28
Circles

21m 3s

Intro
0:00
What are Circles?
0:08
Example: Equidistant
0:17
Radius
0:32
Equation of a Circle
0:44
Example: Standard Form
1:11
Graphing Circles
1:47
Example: Circle
1:56
Center Not at Origin
3:07
Example: Completing the Square
3:51
Example 1: Equation of Circle
6:44
Example 2: Center and Radius
11:51
Example 3: Radius
15:08
Example 4: Equation of Circle
16:57
Ellipses

46m 51s

Intro
0:00
What Are Ellipses?
0:11
Foci
0:23
Properties of Ellipses
1:43
Major Axis, Minor Axis
1:47
Center
1:54
Length of Major Axis and Minor Axis
3:21
Standard Form
5:33
Example: Standard Form of Ellipse
6:09
Vertical Major Axis
9:14
Example: Vertical Major Axis
9:46
Graphing Ellipses
12:51
Complete the Square and Symmetry
13:00
Example: Graphing Ellipse
13:16
Equation with Center at (h, k)
19:57
Horizontal and Vertical
20:14
Difference
20:27
Example: Center at (h, k)
20:55
Example 1: Equation of Ellipse
24:05
Example 2: Equation of Ellipse
27:57
Example 3: Equation of Ellipse
32:32
Example 4: Graph Ellipse
38:27
Hyperbolas

38m 15s

Intro
0:00
What are Hyperbolas?
0:12
Two Branches
0:18
Foci
0:38
Properties
2:00
Transverse Axis and Conjugate Axis
2:06
Vertices
2:46
Length of Transverse Axis
3:14
Distance Between Foci
3:31
Length of Conjugate Axis
3:38
Standard Form
5:45
Vertex Location
6:36
Known Points
6:52
Vertical Transverse Axis
7:26
Vertex Location
7:50
Asymptotes
8:36
Vertex Location
8:56
Rectangle
9:28
Diagonals
10:29
Graphing Hyperbolas
12:58
Example: Hyperbola
13:16
Equation with Center at (h, k)
16:32
Example: Center at (h, k)
17:21
Example 1: Equation of Hyperbola
19:20
Example 2: Equation of Hyperbola
22:48
Example 3: Graph Hyperbola
26:05
Example 4: Equation of Hyperbola
36:29
Conic Sections

18m 43s

Intro
0:00
Conic Sections
0:16
Double Cone Sections
0:24
Standard Form
1:27
General Form
1:37
Identify Conic Sections
2:16
B = 0
2:50
X and Y
3:22
Identify Conic Sections, Cont.
4:46
Parabola
5:17
Circle
5:51
Ellipse
6:31
Hyperbola
7:10
Example 1: Identify Conic Section
8:01
Example 2: Identify Conic Section
11:03
Example 3: Identify Conic Section
11:38
Example 4: Identify Conic Section
14:50
Solving Quadratic Systems

47m 4s

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

21m 16s

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

21m 36s

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

23m 3s

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

22m 43s

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

18m 32s

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

14m 34s

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

48m 30s

Intro
0:00
Pascal's Triangle
0:06
Expand Binomial
0:13
Pascal's Triangle
4:26
Properties
6:52
Example: Properties of Binomials
6:58
Factorials
9:11
Product
9:28
Example: Factorial
9:45
Binomial Theorem
11:08
Example: Binomial Theorem
13:48
Finding a Specific Term
18:36
Example: Specific Term
19:26
Example 1: Expand
24:39
Example 2: Fourth Term
30:26
Example 3: Five Terms
36:13
Example 4: Three Iterates
45:07
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Lecture Comments (7)

1 answer

Last reply by: Dr Carleen Eaton
Sat Nov 7, 2015 5:50 PM

Post by Fadumo Kediye on October 13, 2015

P(x) = 2x^3 + ax^2 +bx + 6 is divided by x + 2, the remainder is -12. If x - 1 is a factor of the polynomial, find the values of a and b.

2 answers

Last reply by: Fadumo Kediye
Tue Oct 13, 2015 11:42 PM

Post by enya zh on September 29, 2012

At about 25:49, you didn't list EVERY POSSIBLE factor. Isn't all the factors 1,(x-2), (x+3), (x-4), (x^2-x-12), (x^2+x-6),(x^2-6x+8),&(x^3-3x^2-10x+24)? 1 and itself would always be factors and I got three additional factors with the degree of two by multiplying the binomial factors.

1 answer

Last reply by: Huseyin Kayahan
Fri Oct 14, 2011 5:45 AM

Post by Huseyin Kayahan on October 14, 2011

In the f(a) example, why the f(2) is equal to remainder?
what is the f(3) with the division method?

Remainder and Factor Theorems

  • Use synthetic substitution to evaluate a polynomial of degree 4 or more for a specific value.
  • Use synthetic division to find the factors of a polynomial of degree 3 or more. Guess and then check your guess by synthetic division. Once you find one factor, use synthetic division to find a factor of the quotient. Keep going until the quotient is quadratic. At this point, use the quadratic formula to factor the quadratic.

Remainder and Factor Theorems

Using synthetic division find f(1) if f(x) = x3 − x2 − x + 1
  • Recall that if you divide f(x) by x − 1 you will get:
  • [f(x)/(x − 1)] = Q + R; the remainder R is f(1).
  • Proceed to do synthetic division, notice that there are no missing terms.
  • 11-1-11
      10-1
     10-1
    0
In this case, f(1) = 0
Using synthetic division find f( − 8) if f(x) = x3 + 7x2 − 13x − 44
  • Recall that if you divide f(x) by x − 8 you will get:
  • [f(x)/(x + 8)] = Q + R; the remainder R is f(8).
  • Proceed to do synthetic division, notice that there are no missing terms.
  • -817-13-44
      -8840
     1-1-5
    -4
In this case, f( − 8) = − 4
Using synthetic division to find f(7) if f(x) = x3 − 9x2 + 12x + 22
  • Recall that if you divide f(x) by x + 7 you will get:
  • [f(x)/(x − 7)] = Q + R; the remainder R is f(7).
  • Proceed to do synthetic division, notice that there are no missing terms.
  • 71-91222
      7-14-14
     1-2-2
    8
In this case, f(7) = 8
Using synthetic division to find f(4) if f(x) = x4 − 4x3 − 7x + 23
  • Recall that if you divide f(x) by x − 4 you will get:
  • [f(x)/(x − 4)] = Q + R; the remainder R is f(4).
  • Proceed to do synthetic division, notice that the square term is missing, add 0 as place holder
  • 41-40-723
      400-28
     100-7
    -5
In this case, f(4) = − 5
Using synthetic division to find f( − 4) if f(x) = 2x4 + 8x3 + 8x + 24
  • Recall that if you divide f(x) by x + 4 you will get:
  • [f(x)/(x + 4)] = Q + R; the remainder R is f( − 4).
  • Proceed to do synthetic division, notice that the square term is missing, add 0 as place holder
  • -4280824
      -800-32
     2008
    -8
In this case, f( − 4) = − 8
Check if (x + 6) is a factor of x3 + 15x2 + 49x − 30
  • Recall that in order for a binomial to be a factor of the given polynomial the Remainder must equal to zero.
  • Divide using synthetic division. Always check for missing terms.
  • -611549-30
      -6-5430
     19-5
    0
Because the remainder is zero, (x + 6) is a factor.
Check if (x + 4) is a factor of 8x3 + 29x2 − 17x − 26
  • Recall that in order for a binomial to be a factor of the given polynomial the Remainder must equal to zero.
  • Divide using synthetic division. Always check for missing terms.
  • -4829-17-26
      -321220
     8-3-5
    -6
Because the remainder is not zero, (x + 4) is not a factor.
Check if (a − 10) is a factor of a5 − 10a4 + 8a − 80
  • Recall that in order for a binomial to be a factor of the given polynomial the Remainder must equal to zero.
  • Divide using synthetic division. Always check for missing terms, in this case, cube and
  • 101-10008-80
      1000080
     10008
    0
Because the remainder is zero, (a − 10) is a factor.
Show that (x − 3) is a factor of x3 − x2 − 14x + 24. Find the other factors.
  • To check that the binomial is a factor, Divide the Polynomial using synthetic division. If the Remainder is 0,
  • then the binomial is a factor. As always, check for any missing terms.
  • To find the other factors, try any methods covered in lessons before.
  • 31-1-1424
      36-24
     12-8
    0
  • (x − 3) is a factor. Now find the other factors by factoring x2 + 2x − 8
  • x2 + 2x − 8 = (x + 4)(x − 2)
  • Factored completely would be
x3 − x2 − 14x + 24 = (x − 3)(x + 4)(x − 2)
Show that (x + 5) is a factor of x3 + 12x2 + 47x + 60. Find the other factors.
  • To check that the binomial is a factor, Divide the Polynomial using synthetic division. If the Remainder is 0,
  • then the binomial is a factor. As always, check for any missing terms.
  • To find the other factors, try any methods covered in lessons before.
  • -51124760
      -5-35-60
     1712
    0
  • (x + 5) is a factor. Now find the other factors by factoring x2 + 7x + 12
  • x2 + 7x + 12 = (x + 4)(x + 3)
  • Factored completely would be:
x3 + 12x2 + 47x + 60 = (x + 5)(x + 4)(x + 3)

*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

Remainder and Factor Theorems

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
  • Remainder Theorem 0:07
    • Checking Work
    • Dividend and Divisor in Theorem
    • Example: f(a)
  • Synthetic Substitution 5:43
    • Example: Polynomial Function
  • Factor Theorem 9:54
    • Example: Numbers
    • Example: Confirm Factor
  • Factoring Polynomials 14:48
    • Example: 3rd Degree Polynomial
  • Example 1: Remainder Theorem 19:17
  • Example 2: Other Factors 21:57
  • Example 3: Remainder Theorem 25:52
  • Example 4: Other Factors 28:21

Transcription: Remainder and Factor Theorems

Welcome to Educator.com.0000

In today's lesson, we will be covering the remainder and factor theorems.0002

Now, today, with the remainder theorem, this is going to be familiar; we used it earlier on, without giving it a name.0007

And we are going to talk about it and put it to a different use today.0014

So, recall that, when we were dividing polynomials, either by long division or synthetic division,0017

a way to check our work was to recall that the dividend equals the quotient times the divisor, plus the remainder.0023

And recall that, if I have, say, 48 divided by 2, this is the dividend, and the term that you are dividing by is the divisor.0042

And of course, the solution is the quotient, and whatever is left over is the remainder.0055

So, sometimes when we did long division, I checked it by multiplying my quotient by the divisor,0059

adding the remainder, and making sure that I came up with the dividend.0066

Now, looking back at all this, this is really just saying the same thing.0070

If the polynomial, f(x), is divided by x - a (here I have the dividend, the polynomial f(x), and the divisor;0075

this is the dividend, f(x); here, the divisor is x - a), I have that the dividend equals the divisor,0084

times...and it tells me that g(x) is the quotient, so times the quotient, plus f(a) (f(a) is the remainder).0096

This is just saying the same thing: if I am working with polynomials, the polynomial equals the quotient times the divisor, plus the remainder.0109

Now, we are going to put this to a different use; and we are going to use it to find f(a).0119

Looking at an example: let's take the polynomial function f(x) = 2x3 + x2 - 3x + 4.0125

And let's say that I wanted to find f(2): well, I could use my substitution method (the usual),0136

where I would just say, "OK, 2 cubed, plus 2 squared" substituting 2 in for x, "minus 3x, plus 4; and this gives me...0145

2 times 23 is 8, plus 4...actually putting in a 2 right here...minus 3 times 2 is 6, plus 4, equals 16 + 4 - 6 + 4."0156

So, this gives me 20 = 2, or 18.0173

OK, that wasn't too bad; but if this had been a polynomial of, say, degree 5 or 6,0179

and I was asked to find something like f(5), that is a lot of arithmetic, working with large numbers.0185

So, there is another way to approach this.0191

What this is saying is that, if I divide a polynomial by x - a, the remainder is f(a).0195

So, instead of finding f(2) by substitution, I could find f(2) a different way: I could say that f(2) equals the remainder when f(x) is divided by x - a.0202

And in this case, I have f(2), so a equals 2.0225

This remainder theorem gives me another method for finding the value of a function at a certain value.0235

So, if I am asked to find f(2), I could just say, "OK, this equals 2x3 + x2 - 3x + 4, divided by x - 2."0246

And if I were to divide all of this out, I would actually find that the remainder is 18.0261

So, I would end up getting a polynomial, f(2), and I would end up getting some quotient;0270

when I did my division, I would get some quotient; plus I would get a remainder after this division.0283

And this remainder is going to be equal to f(2).0291

So, there are two ways for finding the value of a function: if I am asked to find, say, f(2),0298

I can either substitute in, or I can divide the polynomial by x, minus that value (which is 2).0304

And this actually turns out to be 2x2 + 5x + 7, with a remainder of 18.0313

So, if you were to divide this out, you would get this (which is g(x)), and the remainder would be 18.0324

And this remainder of 18 equals f(2).0333

OK, so we are using the remainder theorem to find values for functions.0337

Now, I mentioned that, in order to do this, you are going to have to divide the polynomial by x - a.0344

Long division is a lot of work; so the quicker way to go, since the divisor is in the form x - a constant, is to use synthetic division.0352

And here, we are calling it synthetic substitution only because we are using it to find the value of a polynomial function.0361

It is the same thing: we are using synthetic division, but when we use it for this purpose, we call it synthetic substitution.0368

So, for example, if I was given a polynomial function, f(x) = 4x4 - 2x3 + x - 1,0375

and I was asked to find f(2), I could substitute 2 in here; or I can use this new method.0386

The way I am going to find this is: I am going to take 4x4 - 2x3 + x - 1, and I am going to divide it by x - 2.0397

And then, I am going to find a quotient, plus a remainder; and that remainder is going to equal f(2).0409

Now, reviewing synthetic division (which is much quicker than long division), recall that, in synthetic division,0416

you are going to take this constant here, and put it here, but with the opposite sign.0422

I have a negative 2; I am going to make this a 2.0427

Then, I am going to take these coefficients and put them here.0430

It is very important, before you do that, though, to check for missing terms.0433

So, here, I have x4, x3...I do not have an x2 term; that is missing.0437

So, I am actually going to rewrite this as f(x) = 4x4 - 2x3...and for the missing term,0445

I am going to use a coefficient of 0; that is going to be my placeholder when I do my division...+ x - 1.0452

My coefficients are 4, -2, 0, 1, and -1; with synthetic division, your first step is to bring down that first term.0462

So, this is going to give me 4; then multiply 4 times this divisor, 2, to get 8.0477

Then add: -2 plus 8 gives me 6; multiply again: 6 times 12 gets 12; add: 12...0485

And you see, if I hadn't had this 0 here as a placeholder, I would have gotten a completely different answer.0497

So, it is very important, before you even start writing down the coefficients, to check and see if there are any missing terms.0501

OK, the next step: 12 times 2 gives me 24; add that to 1: 25.0507

25 times 2 gives me 50, plus -1 is 49.0516

Now, what this is giving me is 4x3 (because the variable here is going to have a degree0522

one less than the degree of the dividend, so this is 4x3) + 6x2 + 12x + 25.0531

So, this is the quotient; this last term is the remainder.0541

And this remainder of 49 equals f(2).0553

Again, when you are working with a high-degree polynomial and a large value that you are looking for in the function,0559

then this is a much quicker method than substitution.0568

So again, I handled this by saying I wanted to find f(2), so I am going to divide this polynomial by x - 2,0572

using synthetic division, being careful to include a 0 coefficient as a placeholder for missing terms.0579

I found the quotient and the remainder, and the remainder equals f(2).0585

OK, the factor theorem is a result of the remainder theorem.0594

Now, what this says is that x - a is a factor of the polynomial f(x) if and only if f(a) equals 0.0599

And this makes sense, because if you divide something by its factor, the remainder is going to be 0.0609

Just a very simple case--16: factors include 2 and, say, 4.0616

OK, so if I do 16 divided by 2, it equals 8; the remainder is 0, or 16 divided by 4 equals 4; the remainder is 0.0627

So, I know that x - a is a factor of the polynomial, if and only if the remainder is 0.0639

This tells me something else, too: it tells me that what I have here with a is a 0,0646

because if I have a value of the function, a (let's say I am given a value x = 4 for a polynomial),0653

and if I find that f(4) equals 0, that means that when x equals 4, the value of the function is 0.0662

So, this is a zero of the polynomial, which is very helpful information to have.0669

OK, now, how does this get put into play?0675

What this can help you do is: if you are trying to find the factors, or confirm that something is a factor, you can use synthetic division.0678

For example, let's say I have a polynomial function f(x) = 2x4 - 2x3 - 17x2 + 12x + 9.0686

And let's say that I am asked to determine if x - 3 is a factor of f(x).0700

I may be given a problem to work out, like this; or I may be just asked to factor this.0714

And if I am asked to factor this, I need to start out by just taking some guesses--0719

Is x - 2 a factor? Is x - 3 a factor? Is x - 4 a factor?--and then checking those guesses using synthetic division.0726

So, given f(x), I need to determine if x - 3 is a factor.0733

So, if I divide f(x) by x - 3, if the remainder equals 0, then x - 3 is a factor of f(x).0738

All right, again, using synthetic division is the easiest way to go about it.0760

So, I have x - 3, and I am going to take the opposite sign of -3; I am going to make that a 3 and put that here.0764

Now, I check, and I don't have any missing terms: x4, 3rd, 2nd, 1st, and a constant; I have no missing terms.0770

OK, 2, -2, -17, 12, and 9...0777

Synthetic division: bring that first term down; multiply 3 times 3 to get 6; add 6 and -2 to get 4.0788

Multiply again; 4 times 3 gives me 12; 12 minus 17 is -5; -5 times 3 is -15.0799

-15 plus 12 is -3; -3 times 3 is -9; 9 plus -9 is 0.0809

OK, what this is giving me is my quotient here and my remainder here.0819

So, what this is telling me is: I look up here, and the degree is 4; so the degree of the variable here is actually going to be 3.0826

So, this is telling me that what I have is x - 3, times 2x3 + 4x2 - 5x - 3.0833

And the remainder is 0 here, so x - 3 is a factor of f(x).0852

What I have done now is: I have pulled out this factor, x - 3; I have factored it out of the original; and what is left behind is this.0866

OK, so x - 3 is a factor of f(x); if this remainder had turned out to be anything other than 0, then x - 3 would not have been a factor.0878

So, factoring polynomials: synthetic division can be used to factor polynomials of a degree greater than 2.0888

And we just saw how the factor theorem can help us achieve that.0896

So, if we are working with polynomials that have a greater degree than a quadratic equation, we can use synthetic division to help.0900

For example, if I have something...f(x) = x3 - 5x2 - 2x2 + 24, what I want to do is find one factor.0907

And then, I want to pull that out through synthetic division, just as I did last time.0924

Sometimes you will be given that first factor and just asked to confirm, as I mentioned.0930

Other times, you won't; so then you just have to take guesses.0934

Let me give you an example: is x - 3 a factor? I am just taking a guess.0938

And so, what I am going to do is: I am going to use synthetic division; I am going to put a 3 here, and this coefficient is 1, -5...0947

this x should actually be...let's make that -2x...-2, and 24.0959

OK, so x3, second, first, constant...there are no missing terms.0969

And I am checking to see if x - 3 is a factor; I am going to bring down the 1.0974

1 times 3 is 3; 3 + -5 is -2; -2 times 3 is -6; -6 plus -2 is -8; -8 times -3 is -24.0978

OK, -24 and 24 is 0; my remainder is 0: "Yes," since the remainder is 0 when I divided this polynomial by x - 3.0996

Now, this is degree 3; that tells me that what I have here is going to be degree 2.1012

So, I pulled out this factor, x - 3, and what I have left behind is the quadratic expression x2 - 2x - 8.1019

Not only, now, do I know that this is a factor, but I have something left that I can work with--that I can factor further, using quadratic methods.1037

And in fact, if this were an equation--if I was given that this equals 0 or equals some other number--1046

then I could solve it using either factoring or the quadratic formula.1054

So, just continuing on with the factoring: I have gotten this far; I have pulled out this factor; here is what I have left behind, x - 3.1062

Now, factoring this out, this is just a general trinomial, and I know that it is going to be in this form: (x + something) (x - something), since this is negative.1070

Factors of 8 are 1 and 8, 2 and 4; and I want those to sum up to -2 when one is positive and the other is negative.1081

So, I know that these are too far apart; but if I take 2 and -4, I am going to get -2.1093

So, I am going to factor this out as such; OK, so now this is factored out as far as I can go.1105

Again, synthetic division was very useful, because I started out with this polynomial.1113

And then, I had to just take a guess, and I said, "OK, is x - 3 a factor?"1118

I confirmed that through synthetic division, finding that this remainder is 0.1124

So, this is a factor; so that tells me I have this factor, times what is left behind, which was a quadratic, which I could factor further.1130

If I was wrong, then I would have tried another guess; I would have tried x + 3 or x - 2 or x - 1, until I found one.1139

So, that first step is the longest; if you make the right guess and you get the right factor,1147

and then use synthetic division, then often what you are left with is easier to work with.1151

Applying these concepts to some examples: f(x) = 2x3 - x + 7; I am asked to find f(4).1158

I could find it by substitution; it actually wouldn't be that difficult, just substituting 4 for x wherever the x's appear.1165

But just to practice using the remainder theorem, let's use the remainder theorem and synthetic division, and find f(4) that way.1173

Recall that, if I want to find f(4), what I need to do is divide f(x) by x - 4, and then it is going to give me a quotient plus a remainder.1182

And the remainder is going to equal f(4), according to the remainder theorem.1198

OK, so one thing before I proceed with my division is: I am going to realize that I have a missing term.1204

I have 2x3; I don't have an x2 term, so I am going to represent that with a coefficient of 0.1211

I do have an x term, and then I have my constant.1217

And I am going to divide all that by x - 4.1221

Using synthetic division, I am going to put the 4 out here, and then my coefficients 2, 0, this is -1, and 7,1227

bringing the 2 down; 2 times 4 is 8; 8 plus 0 is 8; 8 times 4 is 32; 32 plus -1 is 31.1237

31 times 4 gives me 124; 124 plus 7 is 131; here is my quotient; here is my remainder.1256

The remainder equals 131; therefore, f(4) equals 131.1270

And you could check this by using the substitution method; and you would find, again, that you got f(4) is 131.1278

Again, I handled this by saying that, if I want to find f(4), I can do that by dividing the function by x - 4, finding the quotient, and finding the remainder.1285

The remainder, when I divide this, is equal to f(4).1295

I use synthetic division, being careful to include 0 for a coefficient for the missing x2 term.1301

I came up with this quotient and a remainder of 131; so, f(4) = 131.1307

Show that x - 2 is a factor of this polynomial, and find the other factors.1319

Recall that, if x - 2 is a factor of this polynomial, then if I divide (I am going to call this f(x)) f(x) by x - 2, the remainder will equal 0.1325

OK, I am supposed to show that this is a factor and go on to find the other factors.1350

So, I am going to do that by using synthetic division.1356

All right, I am dividing by 2; and I do not have any missing terms: cubed, squared, x to the first, constant--no missing terms.1362

So, this is a coefficient of 1, a coefficient of -3, a coefficient of -10, and a coefficient of 24...or actually, the constant is 24.1373

OK, bring down the 1; 1 times 2 is 2; 2 and -3 gives me a -1.1387

-1 times 2 is -2; -2 and -10 is -12; -12 times 2 gives me -24.1396

OK, the remainder equals 0, so x - 2 is a factor of f(x).1414

All right, now this is a degree 3 polynomial; so what I am going to have down here is actually going to be the quadratic equation.1427

That is at 1x2 - 1x - 12.1438

Now, I am supposed to find the other factors; and I always want to remember that x - 2 is a factor.1442

So, x - 2 times this quotient will give me the polynomial back; I don't want to leave this off.1450

But to find the other factors, I am going to have to factor this out farther.1456

So, x - 2 is factored as far as it can be; now, this is going to be a general trinomial in the form (x + something) (x - something), because this is negative.1461

Looking at the factors of 12: the factors of 12 include 1 and 12, 2 and 6, 3 and 4.1479

The middle term here is -1, so I am going to look for factors that, when one of them is negative and the other is positive, add up to -1.1488

-1 is a small number, so I am going to look for factors that are close together; so I am going to focus on this.1497

And I am going to make the larger number negative, because this ends up being negative.1502

-4 + 3 does equal -1; therefore, I am going to make this 3, and this -4.1507

OK, so I did show that x - 2 is a factor, because the remainder is 0.1519

And then, the other factors, in addition to x - 2, are x + 3 and x - 4.1526

This is the complete factorization right here, the one they gave me, and the two other factors.1545

All right, here we are given a function, and we are asked to find f(3),1554

using the remainder theorem, which states that f(3) will be the remainder when I divide this polynomial here by x - 3;1558

I am going to get a quotient plus a remainder, and the remainder is going to equal f(3).1580

OK, using synthetic division: looking here, what I actually have are some missing terms.1586

So, I am going to rewrite this as f(x) = x4...my x3 term is missing,1595

so I will give that a 0 coefficient; I have an x2 term; my x term is missing,1601

so I will give that a 0 coefficient; and I have a constant.1608

So, this is going to give me a coefficient of 1; a coefficient of 0; -2; 0; and 6.1613

Bring down the 1; 1 times 3 is 3; 3 plus 0 is 3; 3 times 3 is 9; 9 plus -2 is 7.1624

7 times 3 is 21; 21 plus 0 is 21; 21 times 3 gives me 63; 63 plus 6 is 69.1639

OK, this is the quotient; this is the remainder.1654

Since the remainder is 69, f(3) equals 69.1660

Now, I could have solved this by substituting 3 in here; but this is actually an easier way to go about it.1669

Again, to find f(3), I am going to divide this polynomial by x - 3; I am going to find my quotient and my remainder; and f(3) equals that remainder of 69.1679

Always be careful to include 0's for coefficients for missing terms.1695

Show that x - 3 is a factor of this polynomial, and find the other factors.1703

If x - 3 is a factor, then when you divide f(x) by x - 3, the remainder will be 0.1711

So, what this is saying is that f(3) equals 0.1726

OK, I am looking, and I do not have any missing terms: cubed, squared, just x to the first, and then a constant.1734

So, I can go ahead with my division.1741

And this is a coefficient of 1; a coefficient of -3; -4; and 12.1745

OK, I am bringing down the 1 to give me 1 times 3 is 3; 3 minus 3 is 0; 0 times 3 is 0; -4 and 0 is -4; -4 times 3 is -12; -12 and 12 is 0.1754

OK, so this is giving me a remainder of 0, so x - 3 is a factor of f(x).1778

I am going to call this f(x).1796

So, I was first asked to show that x - 3 is a factor of this polynomial; and it is, because the remainder is 0.1801

Now, I am supposed to find the other factors.1807

Well, I pulled out this factor, x - 3; so that factor, times what is left over, will give me the polynomial back.1809

What I have here is x3: that means that, after dividing, I am going to be left with1818

1x2 (so x2) + 0x (so I can just leave that out) - 4.1824

All right, so what I have here, x2 - 4, is the difference of two squares: it is in the form a2 - b2.1833

So, I can factor that out to (a + b) (a - b).1841

This gives me (x - 2) (x + 2); this is the complete factorization of this polynomial function.1850

And the factor they gave me is x - 3; the other factors are x - 2 and x + 2.1860

So, I have three factors for this polynomial.1874

That concludes this lesson on Educator.com; and I will see you next time!1877

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