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INSTRUCTORS Carleen Eaton Grant Fraser
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Dr. Carleen Eaton

Dr. Carleen Eaton

Dividing Polynomials

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Table of Contents

I. Equations and Inequalities
Expressions and Formulas

22m 23s

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

20m 15s

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

19m 10s

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

17m 31s

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

17m 14s

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

25m

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

32m 5s

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

14m 46s

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

23m 7s

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

23m 5s

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

31m 5s

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

21m 42s

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

17m 13s

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

23m 53s

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

27m 12s

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

28m 53s

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

11m 34s

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

21m 36s

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

29m 36s

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

33m 13s

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

28m 25s

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

22m 25s

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

22m 32s

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

31m 48s

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

27m 3s

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

19m 53s

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

35m 45s

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

27m 11s

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

22m 48s

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

30m 7s

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

27m 5s

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

19m 29s

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

13m 27s

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

31m 11s

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

22m 30s

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

33m 29s

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

21m 10s

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

31m 21s

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

31m 27s

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

31m 16s

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

34m 30s

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

22m 42s

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

30m 4s

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

20m 46s

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

41m 11s

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

30m 45s

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

31m 27s

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

40m 54s

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

55m 4s

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

57m 13s

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

20m 21s

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

55m 14s

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

35m 58s

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

45m 54s

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

28m 43s

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

25m 23s

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

21m 14s

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

24m 30s

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

32m 42s

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

41m 27s

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

21m 3s

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

46m 51s

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

38m 15s

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

18m 43s

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

47m 4s

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

21m 16s

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

21m 36s

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

23m 3s

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

22m 43s

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

18m 32s

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

14m 34s

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

48m 30s

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

1 answer

Last reply by: Angel La Fayette
Wed Jan 2, 2013 4:11 PM

Post by Angel La Fayette on January 2, 2013

How come the -15 in the dividend was not divided by the 3?

1 answer

Last reply by: Angel La Fayette
Wed Jan 2, 2013 4:11 PM

Post by Daniel Cuellar on October 26, 2012

to fix you mr. Jeff, 6 divided by 3 is 2...

2 answers

Last reply by: Angel La Fayette
Wed Jan 2, 2013 4:06 PM

Post by Jeff Mitchell on March 21, 2011

approx 16:42 into lecture you show
3x^3+12x^2-15x+6 and divide by 3 with result
x^3+4x^2-15X+3 but I believe it should be
x^3+4x^2-5x+3

~Jeff

Dividing Polynomials

  • When dividing a polynomial by a binomial, any missing terms of the dividend must be written explicitly, using a coefficient of 0.
  • In synthetic division, the coefficient of the binomial divisor must be 1. If it is not, the division problem must be rewritten so that it is 1.

Dividing Polynomials

Divide: [(25x5y4 − 15xy3 + 4x3y2)/(5x2y3)]
  • Break down the divident into three terms. Each term must get a copy of the divisor.
  • [(25x5y4)/(5x2y3)] − [(15xy3)/(5x2y3)] + [(4x3y2)/(5x2y3)]
  • Start by simplifying the coefficients of each term if possible.
  • [(5x5y4)/(x2y3)] − [(3xy3)/(x2y3)] + [(4x3y2)/(5x2y3)]
  • Simplify by using properties of exponents.
  • 5x3y − 3x − 1 + [(4xy − 1)/5]
  • Eliminate negative exponents by bringing the variable to denominator, thus making exponents positive
5x3y − [3/x] + [4x/5y]
Divide: [(21x4y4 − 14x2y5 + 7x3y)/(7x6y7)]
  • Break down the divident into three terms. Each term must get a copy of the divisor.
  • [(21x4y4)/(7x6y7)] − [(14x2y5)/(7x6y7)] + [(7x3y)/(7x6y7)]
  • Start by simplifying the coefficients of each term if possible.
  • [(3x4y4)/(x6y7)] − [(2x2y5)/(x6y7)] + [(x3y)/(x6y7)]
  • Simplify by using properties of exponents.
  • 3x − 2y − 3 − 2x − 4y − 2 + x − 3y − 6
  • Eliminate negative exponents by bringing the variable to denominator, thus making exponents positive
[3/(x2y3)] − [2/(x4y2)] + [1/(x3y6)]
Divide [(n3 + 9n2 + 18n + 10)/(n + 1)]
  • Always check that there are no missing terms. If there are any missing terms, always use a place holder such as 0x3. In this case, there are no missing terms, so everything can proceed.
  • Divide n3 by n = n2, then multiply n2 by n and 1 and change the sign of the result (This is the same as adding the opposite as explained in the lesson). Add, then bring down the next term (18n) .
  •   n2  
    n+1n39n218n10
    +−n3−n2 
     08n218n 
  • Divide 8n2 by n = 8n . Multiply 8n by n and 1, change the sign of the result. Add, then bring the next term 10.
  •   n28n 
    n+1n39n218n10
    +−n3−n2
     08n218n
    + −8n2−8n
      010n10
  • Divide 10n by n = 10. Multiply 10 by n and 1, change the sign of the result. Add and notice how there is no reminder.
  •   n28n 
    n+1n39n218n10
    +−n3−n2
     08n218n
    + −8n2−8n
      010n10
    +  −10n−10
n2 + 8n + 10
Divide [(n3 − 9n + 21)/(n + 4)]
  • Always check that there are no missing terms. If there are any missing terms, always use a place holder such as 0x3. In this case, use 0n2 since the square terms is missing.
  • Divide n3 by n = n2, then multiply n2 by n and 4 and change the sign of the result(This is the same as adding the opposite as explained in the lesson). Add, then bring down the next term ( − 9n) .
  •   n2  
    n+4n30n2−9n21
    +−n3−4n2 
     0−4n2−9n 
  • Divide − 4n2 by n = − 4n . Multiply − 4n by n and 4, change the sign of the result. Add, then bring the next term 21.
  •   n2-4n 
    n+4n30n2−9n21
    +−n3−4n2
     0−4n2−9n
    + +4n2+16n
      07n21
  • Divide 7n by n = 7. Multiply 7 by n and 4, change the sign of the result. Add and notice how the remainder is 7.
  •   n2-4n 
    n+4n30n2−9n21
    +−n3−4n2
     0−4n2−9n
    + +4n2+16n
      07n21
       −7n−28
       0−7
n2 − 4n + 7 − [7/(n + 4)]
Divide using Synthetic Division [(x4 + 7x3 + 14x2 + 9x + 9)/(x + 3)]
  • As is the case with long division, always check that there are no missing terms.
  • In this case, there are no missing terms, so you may continue.
  • Also, check that the divisor is in the format (x − r), if it happens to be in the
  • format (ax − r), then everything needs to be divided by a in order to proceed with synthetic division.
  • In this case, x + 3 meets the pattern, therefore, you can divide.
  • Get coeffieients and remember, the number outside is always the opposite sign, in this case, − 3
  • -3171499
          
          
  • Bring the first number down.
  • -3171499
          
     1    
  • Multiply (1) by ( - 3), result goes below 7. Add, result goes below
  • -3171499
      -3   
     14   
  • Multiply (4) by ( - 3), result goes below 14. Add, result goes below
  • -3171499
      -3-12  
     142  
  • Multiply (2) by ( - 3), result goes below 9. Add, result goes below
  • -3171499
      -3-12-6 
     1423 
  • Multiply (3) by ( - 3), result goes below 9. Add, result goes below
  • -3171499
      -3-12-6-9
     14230
  • Remember, the result are the coefficients. The last number is always the remainder, in this case, it's 0.
x3 + 4x2 + 2x + 3
Divide using Synthetic Division [(p4 − 3p3 − 9p2 − 12p − 7)/(p + 1)]
  • As is the case with long division, always check that there are no missing terms.
  • In this case, there are no missing terms, so you may continue.
  • Also, check that the divisor is in the format (x − r), if it happens to be in the
  • format (ax − r), then everything needs to be divided by a in order to proceed with synthetic division.
  • In this case, p + 1 meets the pattern, therefore, you can divide.
  • Get coeffieients and remember, the number outside is always the opposite sign, in this case, − 1
  • -11-3-9-12-7
          
          
  • Bring the first number down.
  • -11-3-9-12-7
          
     1    
  • Multiply (1) by ( − 1), result goes below − 3. Add, result goes below the line
  • -11-3-9-12-7
     1-1   
     1-4   
  • Multiply ( − 4) by ( − 1), result goes below − 9. Add, result goes below the line.
  • -11-3-9-12-7
     1-14  
     1-4-5  
  • Multiply ( − 5) by ( − 1), result goes below − 12. Add, result goes below the line.
  • -11-3-9-12-7
     1-145 
     1-4-5-7 
  • Multiply (7) by ( − 1), result goes below − 7. Add, result goes below the line.
  • -11-3-9-12-7
     1-1457
     1-4-5-70
  • Remember, the result are the coefficients. Start with one power less than the divident. The last number is always the remainder, in this case, it's 0.
x3 − 4x2 − 5x − 7
Divide using Synthetic Division [(p4 − 3p3 + p − 3)/(p − 3)]
  • As is the case with long division, always check that there are no missing terms.
  • In this case, the p2 is missing , therefore, add a place holder in it's place.
  • Also, check that the divisor is in the format (x − r), if it happens to be in the
  • format (ax − r), then everything needs to be divided by a in order to proceed with synthetic division.
  • In this case, p − 3 meets the pattern, therefore, you can divide.
  • Get coeffieients and remember, the number outside is always the opposite sign, in this case, + 3
  • +31-301-3
          
          
  • Bring the first number down.
  • +31-301-3
          
     1    
  • Multiply (1) by (3), result goes below − 3. Add, result goes below the line
  • +31-301-3
      3   
     10   
  • Multiply (0) by (3), result goes below 0. Add, result goes below the line.
  • +31-301-3
      30  
     100  
  • Multiply (0) by (3), result goes below 1. Add, result goes below the line.
  • +31-301-3
      300 
     1001 
  • Multiply (1) by ( + 3), result goes below − 3. Add, result goes below the line.
  • +31-301-3
      3003
     10010
  • Remember, the result are the coefficients. Start with one power less than the divident. The last number is always the remainder, in this case, it's 0.
p3 + 0p2 + 0p + 1 = p3 + 1
Divide using Synthetic Division [(2b4 − 6b3 + 5b − 17)/(b − 3)]
  • As is the case with long division, always check that there are no missing terms.
  • In this case, the b2 is missing , therefore, add a place holder in its place.
  • Also, check that the divisor is in the format (x − r), if it happens to be in the
  • format (ax − r), then everything needs to be divided by a in order to proceed with synthetic division.
  • In this case, b − 3 meets the pattern, therefore, you can divide.
  • Get coeffieients and remember, the number outside is always the opposite sign, in this case, + 3
  • +32-605-17
          
          
  • Bring the first number down.
  • +32-605-17
          
     2    
  • Multiply (2) by (3), result goes below − 6. Add, result goes below the line
  • +32-605-17
      6   
     20   
  • Multiply (0) by (3), result goes below 0. Add, result goes below the line.
  • +32-605-17
      60  
     200  
  • Multiply (0) by (3), result goes below 5. Add, result goes below the line.
  • +32-605-17
      600 
     2005 
  • Multiply (5) by ( + 3), result goes below − 17. Add, result goes below the line.
  • +32-605-17
      60015
     2005−2
  • Remember, the result are the coefficients. Start with one power less than the divident. The last number is always the remainder, in this case, it's − 2.
  • 2b3 + 0b2 + 0b + 5 − [2/(b − 3)] = 2b3 + 5 − [2/(b − 3)]
2b3 + 5 − [2/(b − 3)]
Divide using Synthetic Division [(4n3 − 2n2 − 4n)/(4n − 2)]
  • As is the case with long division, always check that there are no missing terms.
  • In this case, the constant term is missing , therefore, add a place holder in its place.
  • Also, check that the divisor is in the format (x − r), if it happens to be in the
  • format (ax − r), then everything needs to be divided by a in order to proceed with synthetic division.
  • In this case, 4n − 2 does not meets the pattern, therefore, you have to make it by dividing everything by 4.
  • [(4n3)/4] − [(2n2)/4] − [4n/4] + [0/4] ÷[4n/4] − [2/4]
  • Simplify as much as possible
  • n3 − [(1n2)/2] − n + 0 ÷n − [1/2]
  • Get coeffieients and remember, the number outside is always the opposite sign, in this case, + [1/2]
  • [1/2]1−[1/2]-10
         
         
  • Bring the first number down.
  • [1/2]1−[1/2]-10
         
     1   
  • Multiply (1) by (1/2), result goes below − 1/2. Add, result goes below the line
  • [1/2]1−[1/2]-10
      [1/2]  
     10  
  • Multiply (0) by (1/2), result goes below − 1. Add, result goes below the line.
  • [1/2]1−[1/2]-10
      [1/2]0 
     10-1 
  • Multiply ( − 1) by (1/2), result goes below 0. Add, result goes below the line.
  • [1/2]1−[1/2]-10
      [1/2]0−[1/2]
     10-1
    −[1/2]
  • Remember, the result are the coefficients. Start with one power less than the divident. The last number is always the remainder, in this case, it's − 1/2.
  • Using some algebra one to reduce the complex fraction [([1/2])/(n − [1/2])] you get:
  • n2 − n − [([1/2])/(n − [1/2])] = n2 − n − [([1/2])/([2n/2] − [1/2])]
  • n2 − n − [([1/2])/([2n/2] − [1/2])] = n2 − n − [([1/2])/([(2n − 1)/2])]
  • n2 − n − [([1/2])/([(2n − 1)/2])] = n2 − n − ( [1/2] ÷[(2n − 1)/2] )
  • n2 − n − ( [1/2] ÷[(2n − 1)/2] ) = n2 − n − ( [1/2]*[2/(2n − 1)] )
  • n2 − n − ( [1/2]*[2/(2n − 1)] ) = n2 − n − [1/(2n − 1)]
n2 − n − [1/(2n − 1)]
Divide using Synthetic Division [(3r3 − 7r2 − 8r + 14)/(3r − 4)]
  • As is the case with long division, always check that there are no missing terms.
  • In this case, the constant term is missing , therefore, add a place holder in its place.
  • Also, check that the divisor is in the format (x − r), if it happens to be in the
  • format (ax − r), then everything needs to be divided by a in order to proceed with synthetic division.
  • In this case, 3r − 4 does not meets the pattern, therefore, you have to make it by dividing everything by 3.
  • [(3r3)/3] − [(7r2)/3] − [8r/3] + [14/3] ÷[3r/3] − [4/3]
  • Simplify as much as possible
  • r3 − [(7r2)/3] − [8r/3] + [14/3] ÷r − [4/3]
  • Get coeffieients and remember, the number outside is always the opposite sign, in this case, + [4/3]
  • [4/3]1−[7/3]−[8/3][14/3]
         
         
  • Bring the first number down.
  • [4/3]1−[7/3]−[8/3][14/3]
         
     1   
  • Multiply (1) by (4/3), result goes below − 7/3. Add, result goes below the line
  • [4/3]1−[7/3]−[8/3][14/3]
      [4/3]  
     1 −[3/3]  
  • Multiply ( − 1) by (4/3), result goes below − 8/3. Add, result goes below the line.
  • [4/3]1−[7/3]−[8/3][14/3]
      [4/3]−[4/3] 
     1 −[3/3] = −1−[12/3]=−4 
  • Multiply ( − 4) by (4/3), result goes below 14/3. Add, result goes below the line.
  • [4/3]1−[7/3]−[8/3][14/3]
      [4/3]−[4/3]−[16/3]
     1 −[3/3] = −1−[12/3]=−4
    −[2/3]
  • Remember, the result are the coefficients. Start with one power less than the divident. The last number is always the remainder, in this case, it's − 2/3.
  • Using some algebra one to reduce the complex fraction [([1/2])/(n − [1/2])] you get:
  • r2 − r − 4 − [([2/3])/(r − [4/3])] = r2 − r − 4 − [([2/3])/([3r/3] − [4/3])]
  • r2 − r − 4 − [([2/3])/([3r/3] − [4/3])] = r2 − r − 4 − [([2/3])/([(3r − 4)/3])]
  • r2 − r − 4 − [([2/3])/([(3r − 4)/3])] = r2 − r − 4 − ( [2/3] ÷[(3r − 4)/3] )
  • r2 − r − 4 − ( [2/3] ÷[(3r − 4)/3] ) = r2 − r − 4 − ( [2/3]*[3/(3r − 4)] )
  • r2 − r − 4 − ( [2/3]*[3/(3r − 4)] ) = r2 − r − 4 − [2/(3r − 4)]
r2 − r − 4 − [2/(3r − 4)]

*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

Dividing Polynomials

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
  • Dividing by a Monomial 0:13
    • Example: Numbers
    • Example: Polynomial by a Monomial
  • Long Division 2:28
    • Remainder Term
    • Example: Dividing with Numbers
    • Example: With Polynomials
    • Example: Missing Terms
  • Synthetic Division 11:44
    • Restriction
    • Example: Divisor in Form
  • Divisor in Synthetic Division 15:54
    • Example: Coefficient to 1
  • Example 1: Divide Polynomials 17:10
  • Example 2: Divide Polynomials 19:08
  • Example 3: Synthetic Division 21:42
  • Example 4: Synthetic Division 25:09

Transcription: Dividing Polynomials

Welcome to Educator.com.0000

Today, we are going to be talking about dividing polynomials, starting out with a review of techniques learned in Algebra I,0002

and then going on to learn a new technique called synthetic division.0008

First, we are discussing dividing a polynomial by a monomial.0013

The technique is to divide each term of the polynomial by the monomial.0017

And if you think about just dividing with regular numbers, if you have something like this,0023

you could handle it by saying, "15 divided by 3, plus 6 divided by 3, plus 24 divided by 3," and splitting that up.0030

This would give you 5; 6 divided by 3 is 2; and then, 24 divided by 3 is 8; adding this up gives you 15.0041

You may also have done this and added 15, 6, and 24; and those add up to 45/3, to get 15.0054

So, these two are equivalent; and that is what allows you to handle dividing a polynomial by a monomial, by dividing each term by the monomial.0068

For example, 10x4 + 8x3 - 12x, all divided by 2x:0078

looking at this up here as my example of how to handle it, I am going to say,0088

"OK, this is equivalent to 10x4 divided by 2x, plus 8x3 divided by 2x, minus 12x divided by 2x.0092

And this is a step you might do in your head, or you might write it out.0106

10 divided by 2 is 5; x4 divided by x would be the same as saying x4 - 1, or x3.0111

8 divided by 2 gives me 4; x3 divided by x is x2.0122

Here, I have -12 divided by 2 (is 6); and the x's cancel.0129

My result is 5x3 + 4x2 - 6.0135

And I handled that by dividing each term in the polynomial by the monomial, separately.0140

OK, now when you are working with dividing a polynomial by another polynomial, one technique is long division.0148

And we talked about this in Algebra I, and you can also review those lectures; and I will review it here, as well.0154

Just as with regular division with numbers (long division) you might end up with a remainder.0161

So, first, just reviewing long division (which I am sure you know well, but just to think about the steps you are taking):0166

you probably do this so much that you don't even think about the steps,0173

but you want to realize what you are doing each step of the way,0175

so that you can apply it when you are dividing polynomials.0180

If you were asked to divide something like 513 by 2 by long division, think about what you would be doing.0184

First, you divide with the first term, using the first term in the divisor: 5 divided by 2.0192

OK, so that is going to give me 2.0202

Next, I am going to multiply--I am going to multiply 2 by the divisor 2 to get 4.0204

And then, I am going to subtract that product: 2 times 2 is 4, so I am going to subtract that--that is going to give me 1.0213

Next, I will bring down the next number; in the case of polynomials, I will bring down the next term.0224

OK, 2 goes into 11 five times; this gives me 5 times 2 (is 10): 11 minus 10 is 1; now I am bringing down this 3.0235

2 goes into 13 six times, and this is going to give me 12, because now I am multiplying and then subtracting; and I have a remainder of 1.0255

Remember that you can always check your answer by realizing that the dividend equals the divisor, times the quotient, plus the remainder.0265

Looking at this with these numbers, my dividend here is 513; so it equals the divisor (which is 2), times 256, plus the remainder of 1, which equals 513.0283

So, that checks out.0297

Now, with polynomials, it's the same concept.0298

Look at this example, which is going to be 5x2 + 4x - 7, all divided by x + 3.0301

Using long division: it is the same steps that I took over here.0316

I am going to start out with x and divide that into 5x2; I have 5x2 divided by x;0324

and you can see that this is going to give me 5x, because one of the x's will cancel.0331

OK, so that gives me 5x; now, my next step is to multiply 5x times x, which gives me 5x2.0337

5x times 3 is 15x; now, I divided; I multiplied; the next step is to subtract.0350

The 5x's cancel out; 4x - 15x is -11x.0362

The next step: bring down the next number--and here, that would be -7, the next term.0370

OK, so I go back up; I was down here--I go back to #1 again and divide.0378

-11x divided by x; the x's cancel, so that gives me -11.0383

So, I am going to put that up here; I divided; now I need to multiply.0391

-11 times x is -11x; -11 times 3 gives me -33, minus (I subtract--subtracting is the same as adding the opposite,0395

so I am going to add, and the opposite of these two would be positive terms)...-11x and 11x cancels out.0409

And then, I have -7 + 33, which is going to give me 26; and that is my remainder.0417

So, my answer is the quotient, 5x - 11, plus the remainder of 26.0424

Now, to check this out, I am going to say, "OK, the dividend equals the divisor, times the quotient, plus the remainder."0429

Let's make sure this checks out: that is 5x2 - 11x + 15x - 33 (just using the FOIL method) + 26.0446

Simplify to get 5x2...-11x + 15x gives me 4x; -33 + 26 is -7.0462

And this does check out; so again, this is really just using the same techniques as you used for long division with numbers.0470

Now, one thing to be aware of is that sometimes there are missing terms.0477

And if that is the case, you need to use a coefficient of 0 and represent those terms when you are dividing.0483

Now, if you think about it, when we have numbers like, say, 107, we have a 0 here as a placeholder.0490

When we are doing long division with polynomials, we need to do the same thing.0500

For example, let's say you were asked to divide 3x3 + 6x - 4: you were asked to divide that by x + 2.0505

If you look here, there is a missing term: I have an x3 term, but I have no x2 term.0517

And then, I have an x and a constant.0524

So, I have a missing x2 term; and really, I can represent that x2 term by giving it a coefficient of 0.0525

And I need to do that before I divide.0537

So, if I have x + 2, that is my divisor; my dividend--what I am going to write here--is0539

3x3 + (my missing term) 0x2 + 6x - 4.0546

And then, I am going to go about long division like I usually do.0555

So, I take 3x3, and I divide that by x; and that is going to give me 3x2,0559

because 3x3 divided by x is going to give me 3x2.0566

OK, so I divided; the next step is to multiply: 3x2 times x is 3x3; 3x2 times 2 is 6x2.0574

The next step--subtract: these cancel out; this is 0x2 - 6x2.0584

And you can see how, if you didn't have that missing term in there, you would get a completely different answer.0592

OK, so now I have -6x2; I am going to bring down this next term, 6x.0599

OK, now x goes into -6x2 -6x times; so I am going to put -6x up here.0605

I divided; now multiply: -6x times x gives me -6x2; -6x times 2 is -12x.0620

Now, I am subtracting; but remember, that is the same as adding the opposite--I am going to change these signs.0629

These cancel out; and then, I end up with 6x + 12x, which is 18x.0636

Bring down the next term, -4.0644

x goes into 18x 18 times, so up here, I am going to have an 18.0648

18 times x is 18x; 18 times 2 is 36; now I subtract.0659

18x - 18x: that cancels; this negative applies to here as well, so I have -4 minus 36 to get -40.0670

So, the remainder equals -40; and so, my answer is 3x2 - 6x + 18, with a remainder of -40.0679

Again, the key point here was to make sure that, when you notice that there is a missing term0693

(the x2 term is missing) to represent that using the coefficient of 0.0699

OK, there is another way to divide that is much faster and easier than long division.0704

But it is only applicable to dividing a polynomial by a binomial.0709

So, when you have those cases, synthetic division is an excellent method to use.0715

Now, there is another restriction, and that is that the divisor (which, in this case, is the binomial) must be in the form x - r, where r is a constant.0721

So, the divisor has to be in this form; if it is not in that form, you have to get it into that form.0731

OK, let's take a look at an example where the divisor is already in this form, to keep it simple for now.0738

3x3 - 2x2 + 7x + 4; if I am asked to divide that by x - 2,0747

and it is in this form of x minus a constant, I could use long division.0757

But you will see that this is a much faster method.0763

So, first, draw a symbol like this; now, you put the r term right here (in this case, it is 2).0766

And it says - 2, but you actually use the opposite sign; if this were to say x + 3, then I would put a negative here.0777

So, this is x - 2; I use the opposite sign, so I am going to put a 2 right here.0785

Now, what I write in here are the coefficients of the dividend: 3, -2, 7, and 4.0792

And just as with long division, if there is a missing term (let's say my x2 term was missing),0805

I would have represented that here with a coefficient of 0--the same idea as with long division.0810

OK, so here I have the constant; here I just have the coefficients from up here.0815

The first step is to bring down that first coefficient and just put it here.0821

OK, the second step is to multiply 2 times 3; I am going to multiply 2 times 3, and my result will be 6.0826

All I am going to do is add this to -2; -2 + 6--that is going to give me 4.0839

Then, I repeat that process: I am going to multiply -2 by 4 to get 8, and I am going to add: 8 + 7 is 15.0849

I am going to repeat that again: -2 times 15 is going to give me 30; and I am going to add again: 4 + 30 is 34.0865

Now, what do these numbers represent? Well, they represent the quotient and the remainder.0876

And to write them out, what you are going to do is look at the dividend.0888

And the quotient is going to be a degree one less than the degree of the dividend.0892

So, the degree of the dividend was 3; so the degree of the quotient is going to be 2.0897

I am going to write this out as 3x2 + 4x + 15, with a remainder of 34.0901

And you can see how much quicker and easier this is than long division.0915

Again, write the constant from the divisor here; write the coefficients here, making sure to use 00919

if you have a missing term, using 0 as the coefficient; bring down the first term.0924

Then, multiply the divisor constant by this number and get a product.0929

Add that to the next column; then multiply 2 times this number, 4, to get 8; add to this column, 15.0937

2 times 15 is 30; add to this column to get 34, which is the remainder.0946

OK, as I mentioned, the divisor must be in the form x - r; if the coefficient of x is not 1,0954

you need to rewrite it so that the coefficient is 1 in order to use this method.0962

For example, if I was given 3x3 + 12x2 - 15x + 6, divided by 3x - 9,0968

I can see that this coefficient is not 1; in order to make it 1, I need to divide each term0981

in both the dividend and the divisor by this coefficient.0987

So, I am going to divide each term by 3; and that is going to give me x3 + 4x2 - 15x + 2, divided by x - 3.0992

Now, I can go about synthetic division in the usual way.1007

And as you will see here, I got lucky, and dividing things by 3 still kept everything as integers.1010

However, it is very possible that, when you divide by this number, you are going to end up with some fractions.1016

So, that is a drawback, and it makes it more complicated; but it is necessary to do this in order to use synthetic division.1022

OK, first we are going to practice dividing a polynomial by a monomial.1030

And you will recall that the technique is to divide each term in the polynomial by the monomial.1035

So, I am rewriting this, dividing each term by the monomial.1042

So, 20 divided by 4 gives you 5; the x's cancel out; and then, y4 divided by y2...this is 4 - 2...is y2, 5y2.1059

12 divided by 4 (and there is a negative sign in front of that) gives me 3.1073

x to the first minus 2...this is going to give me x to the -1, and I am going to have to take care of that in a minute to put it in my final form.1080

But for right now, we will leave it like that.1091

y3 divided by y2...3 - 2 just gives me y, plus...I'll just leave this as 5/4...1093

x3 divided by x2 is x; and then here, I have y, which is really y1,1103

divided by y2, is 1 - 2; and this is y-1.1113

Now, remember: you want to simplify things, and they are not in simplest form if you have negative powers.1116

So, recalling that rule that a-n equals 1/an, I can simplify by moving this x-1 to the denominator.1122

And I can do the same thing here: x stays up here; y moves to the denominator.1136

And this is my answer: dividing a polynomial by a monomial, and then simplifying.1143

OK, this next problem, dividing a polynomial by a monomial, could be done by synthetic division, because it is in this form x - r.1150

But just to get a little more practice on long division, let's do long division on this one.1159

x goes into 3x3...this becomes 2, so this is 3x2; so it goes into it 3x2 times.1172

So, I divided; and now I am going to multiply: 3x2 times x is 3x3;1187

3x2 times -3 is -9x2.1193

Now, I am subtracting; and that is the same as adding the opposite, so this becomes negative; this becomes positive.1197

These cancel out, so I have -2x2 + 9x2 is 7x2.1205

Bring down the next term, and then divide again.1213

I have 7x2 divided by x equals 7x; 7x times x; 7x, -3 gives me -21x.1217

Subtract (which means I am adding the opposite): this becomes a negative; this one becomes a positive.1233

This cancels; 4x + 21x is 25x; now, divide again: 25x divided by x...the x's cancel; that is 25.1240

So, I am going to have 25 up here.1255

Multiply 25 times x; that is 25x; and bring down this next term; I have -8 there.1257

So, 25 times -3 is going to give me -75; I am going to subtract, which is adding the opposite.1269

So, that is -25 and positive 75; these cancel; -8 + 75 is 67, and that is my remainder.1280

So, the answer is 3x2 + 7x + 25, with a remainder of 67.1289

That was long division; now, this next example specifies to divide using synthetic division.1298

And I am checking, and the divisor is in the form x - r; so I can go ahead and do it without any further manipulation of the expression.1307

Recall: in synthetic division, set it up as follows: you are going to put the constant here, with the opposite sign.1321

This is -4; I am going to make it a 4.1329

Now, before I proceed with putting the coefficients in, it is very important to check for missing coefficients.1332

And I am looking, and I have y4; I do not have y3; I have y2, y, and a constant.1339

So, I am going to rewrite this using 0 for my coefficient with the missing term.1347

I have a missing y3 term, and I am going to rewrite it; and the coefficient is 0 (to use that as a placeholder).1358

So, this is really what I am going to do; OK.1369

Now, I am going to use these coefficients: I have 5, 0, -3, 2, and then my constant, -8.1372

The first step is just to bring down that first term, 5; the second step is to multiply.1386

4 times 5 is 20; after multiplying, add: 0 and 20 is 20; multiply again: 4 times 20 is 80; -3 and 80 is 77.1392

4 times 77--if you calculated that out, you would find that it is 308; 308 + 2 is 310.1417

4 times 310--if you work that out, you will find that it is 1240; 1240 and -8 is going to give you 1232.1431

So again, write the coefficients here, being careful to realize you have a missing term,1444

so you need to represent the coefficient for that missing term as 0.1448

Bring down the first term, and then multiply and add that product to the next column.1452

Find the sum; multiply; add the product to the next column; find the sum; and continue on.1460

Now, for my quotient: this is the quotient; this is going to be the remainder.1467

The quotient is going to have a degree one less than the degree of the dividend;1479

so the degree of the dividend is 4, so the degree of the quotient is going to be 3;1483

so I am going to write this out as 5y3 + 20y2 + 77y + 310;1488

the remainder is 1232, which is a big remainder, but this is correct.1498

In this example, we also are asked to do synthetic division; and I check, and I have a couple of things going on here.1510

I have a missing coefficient, and this is not in the correct form of x - r.1518

So, dealing with the missing term--I have a missing term, which I am going to represent with a coefficient of 0.1526

So, first addressing the missing term: I am missing a z2 term; I have 6z4 - 8z3.1533

Since I have no z2 term, I am going to use the coefficient of 0 for that: 0z2 - 4z + 8.1543

OK, that is taken care of; now, the other problem I have is that this is not in the correct form.1552

In order to have this z have a coefficient of 1, I need to divide all the terms in the divisor and the dividend by 2, so I am going to do that.1558

So, I divide by 2--divide each term by 2 to get the form...I am going to say z instead of x...the form z - r.1569

OK, so 6z4: dividing that by 2 is going to give me 3z4.1590

8z3 divided by 2 is -4z3; 0z2 divided by 2 is 0z2.1596

-4z divided by 2 is -2z; 8 divided by 2 is 4; so far, pretty good.1611

2z divided by 2 gives me z, which is what I wanted; now, here, when I divide -1 by 2, I am going to get a fraction.1620

And it makes it more difficult to work with, but you can still do the synthetic division.1628

Now, I am ready to set this up: here, I am going to take -1/2; I am going to take the opposite sign and write it here; that is 1/2.1634

Then, I am going to put the coefficients here--my new coefficients, after dividing: 3, -4, 0, -2, and 4.1644

OK, bring down the 3 and multiply 3 by 1/2; 3 times 1/2 is just 3/2.1659

1/2 times 3 is 3/2; now, you might need to work out the arithmetic on the side, and that is fine.1670

I have -4 and 3/2; -4 is equal to -8/2; I want to get a common denominator.1677

Adding that to 3/2 is going to give me -5/2, so this is -5/2.1685

Now, multiplying 1/2 by -5/2 is going to give me -5/4; this times this is going to give me -5/4.1695

0 and -5/4 is just -5/4; now, I have to multiply 1/2 by -5/4, and this is going to give me -5/8; I have -5/8 here.1709

I want to get a common denominator; and so, I am going to do -2 times 8, which is going to give me -16/8.1724

So, this is -16/8 + -5/8; and it is just -16 and -5, is -21/8; I am combining these to get -21/8.1737

OK, now 1/2 times -21/8 equals -21/16; this times this is -21/16.1753

I have to add that to 4; so I want to get a common denominator: 4 times 16 is going to give me 64, so 4 = 64/16.1771

So, I want to take 64/16 - 21/16; and that is going to be...64 - 21 is 43/16.1788

OK, this was kind of messy to do; but synthetic division does work, and will end up giving you the correct answer.1801

So, what you need to look at here is that this has a degree of 4, so this is actually going to be a degree of 3.1809

So, rewriting this up here, my quotient is going to be 3z3 - 5/2z2 - 5/4z - 21/8.1816

And I have a remainder of 43/16.1841

So, two things to notice: we had a missing term here--the z2 term was missing--1846

so I had to use a coefficient of 0; and the second is that this was not in this form.1852

So, I had to divide every term in the numerator and the denominator, the divisor and the dividend, by 2, and then proceed as usual with synthetic division.1856

That concludes this lesson on dividing polynomials for Educator.com.1865

And I will see you again soon!1871

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