INSTRUCTORS Carleen Eaton Grant Fraser Eric Smith

Eric Smith

Adding & Subtracting Polynomials

Slide Duration:

Section 1: Properties of Real Numbers
Basic Types of Numbers

30m 41s

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

19m 26s

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

12m 6s

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

18m 52s

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

12m 22s

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

28m 52s

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

12m 2s

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

28m 41s

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

24m 26s

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

22m 55s

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

27m 58s

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

20m 36s

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

21m 24s

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

16m 12s

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

25m 54s

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

20m 1s

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

19m 40s

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

24m 34s

Intro
0:00
Objectives
0:12
Applications of Systems of Equations
0:30
Word Problems
1:31
Example 1
2:17
Example 2
7:55
Example 3
13:07
Example 4
17:15
Section 6: Inequalities
Solving Linear Inequalities in One Variable

17m 13s

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

16m 13s

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

14m 12s

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

17m 7s

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

15m 33s

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

21m 13s

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

44m 51s

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

18m 33s

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

25m 7s

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

44m 56s

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

28m 27s

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

21m 44s

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

30m 9s

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

30m 14s

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

23m 38s

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

29m 27s

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

16m 47s

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

29m 4s

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

26m 53s

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

21m 42s

Intro
0:00
Objectives
0:07
Polynomial Inequalities
0:30
Solving Polynomial Inequalities
1:20
Example 1
2:45
Polynomial Inequalities Cont.
5:12
Larger Polynomials
5:13
Positive or Negative Intervals
7:16
Example 2
9:01
Example 3
13:53
Section 10: Rational Equations
Multiply & Divide Rational Expressions

26m 41s

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

20m 24s

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

18m 23s

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

16m 24s

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

18m 54s

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

20m 20s

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

27m 4s

Intro
0:00
Objectives
0:10
Variation and Proportion
0:34
Variation
0:35
Inverse Variation
1:01
Direct Variation
1:10
Setting Up Proportions
1:31
Example 1
2:27
Example 2
5:36
Variation and Proportion Cont.
8:29
Inverse Variation
8:30
Example 3
9:20
Variation and Proportion Cont.
12:41
Constant of Proportionality
12:42
Example 4
13:59
Variation and Proportion Cont.
16:17
Varies Directly as the nth Power
16:30
Varies Inversely as the nth Power
16:53
Varies Jointly
17:09
Combining Variation Models
17:36
Example 5
19:09
Example 6
22:10
Section 11: Radical Equations
Rational Exponents

14m 32s

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

15m 12s

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

17m 22s

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

19m 24s

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

15m 5s

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

29m 16s

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

### Adding & Subtracting Polynomials

• A polynomial is a term or a finite sum of terms in which all variables have whole number exponents and no variables appear in the denominator.
• Polynomials are classified by how many terms they have, and the largest power present.
• To evaluate a polynomial we substitute a value into all variables, and then simplify.
• To add or subtract polynomials together, we add or subtract the like terms of each of them.
• When adding or subtracting it is often helpful to line up the like terms.

### Adding & Subtracting Polynomials

( 3x2 − 5x − 8 ) + ( 5x4 − 9x2 + x )
• ( 3x2 − 9x2 ) + ( − 5x + x ) + 5x4 − 8
• − 6x2 − 4x + 5x4 − 8
5x4 − 6x2 − 4x − 8
( 6y3 + 3y2 − 7y + 2 ) + ( 4y2 + 10y )
• ( 3y2 + 4y2 ) + ( − 7y + 10y ) + 6y3 + 2
• 7y2 + 3y + 6y3 + 2
6y3 + 7y2 + 3y + 2
( x4 + x3 − x2 + 2x ) + ( 12x4 − 3x3 + 6x + 14 )
• ( x4 + 12x4 ) + ( x3 − 3x3 ) + ( 2x + 6x ) − x2 + 14
• 13x4 − 2x3 + 8x − x2 + 14
13x4 − 2x3 − x2 + 8x + 14
Subtract:
( 10x2 − 4x4 + 5x3 − 18 ) − ( 5x3 + 9x − x2 )
• ( 10x2 − 4x4 + 5x3 − 18 ) + ( − 5x3 − 9x + x2 )
• ( 10x2 + x2 ) + ( 5x3 − 5x3 ) + ( − 4x4 ) + ( − 9x ) − 18
• ( 11x2 ) + 0 − 4x4 − 9x − 18
− 4x4 + 11x2 − 9x − 18
Subtract:
( 3d2 − 4d3 + 6d ) − ( 7d + 5d3 − 11d2 )
• ( 3d2 − 4d3 + 6d ) + ( − 7d − 5d3 + 11d2 )
• ( 3d2 + 11d2 ) + ( − 4d3 − 5d3 ) + ( 6d − 7d )
• 13d2 − 9d3 − d
− 9d3 + 13d2 − d
Subtract:
( 4g3 + 2g5 − 8g2 ) − ( 6g4 + 14g2 − 18g3 )
• ( 4g3 + 2g5 − 8g2 ) + ( − 6g4 − 14g2 + 18g3 )
• ( 4g3 + 18g3 ) + ( − 8g2 − 14g2 ) + 2g5 − 6g4
• 22g3 − 22g3 + 2g5 − 6g4
2g5 − 6g4 + 22g3 − 22g3
( 3m2 − m + 5 ) − ( 6m − 12m2 ) + ( 20m2 − 10m − 1 )
• ( 3m2 − m + 5 ) + ( − 6m + 12m2 ) + ( 20m2 − 10m − 1 )
• ( 3m2 + 12m2 + 20m2 ) + ( − m − 6m − 10m ) + ( 5 − 1 )
35m2 − 17m + 4
( 4s3 − 16s4 + 3s ) + ( 7s2 − 9s3 + s ) − ( 11s4 − 2s2 + 8s3 + 15 )
• ( 4s3 − 16s4 + 3s ) + ( 7s2 − 9s3 + s ) + ( − 11s4 + 2s2 − 8s3 − 15 )
• ( 4s3 − 9s3 − 8s3 ) + ( − 16s4 − 11s4 ) + ( 3s + s ) + ( 7s2 + 2s2 ) − 15
• − 13s3 − 27s4 + 4s + 9s2 − 15
− 27s4 − 13s3 + 9s2 + 4s − 15
( 3c2 − 18c3 + 14c ) − ( 12c + 15c2 − 16c ) + ( 7c3 − 10c2 )
• ( 3c2 − 18c3 + 14c ) + ( − 12c − 15c2 + 16c ) + ( 7c3 − 10c2 )
• ( 3c2 − 15c2 − 10c2 ) + ( − 18c3 + 7c3 ) + ( 14c − 12c + 16c )
• − 22c2 − 9c3 + 18c
− 9c3 − 22c2 + 18c
( 5t5 − 8t3 + 2t − 9 ) − ( 4t2 − 7t + 13 + 21t − t2 ) + ( 9t3 + t5 − 17 + 16t )
• ( 5t5 − 8t3 + 2t − 9 ) + ( − 4t2 + 7t − 13 − 21t + t2 ) + ( 9t3 + t5 − 17 + 16t )
• ( 5t5 + t5 ) + ( − 8t3 + 9t3 ) + ( 2t + 7t − 21t + 16t ) + ( − 4t2 + t2 ) + ( − 9 − 13 − 17 )
• 6t5 + t3 + 4t − 3t2 − 39
6t5 + t3 − 3t2 + 4t − 39

*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.

### Adding & Subtracting 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
• Objectives 0:07
• Adding and Subtracting Polynomials 0:25
• Terms
• Coefficients
• Like Terms
• Polynomials
• Monomials, Binomials, Trinomials, and Polynomials
• Degrees
• Evaluating Polynomials
• Adding and Subtracting Polynomials Cont. 9:25
• Example 1 11:48
• Example 2 13:00
• Example 3 14:41
• Example 4 16:15

### Transcription: Adding & Subtracting Polynomials

Welcome back to www.educator.com.0000

In this lesson we are going to take a look at how you can add and subtract polynomials.0002

Before we get too far though or have to say what polynomials are and how we can classify them.0009

I think we will finally get into adding and subtracting them.0015

To watch for along the way, I will cover how you can evaluate polynomials for several different values.0018

Recall some vocabulary that we had earlier.0029

A term is a piece that is either connected using addition or subtraction.0033

In this little example down below, this expression I have four terms.0040

A coefficient is the number in front of the variable.0051

The coefficient of this first term would be a 4 and I have a coefficient of 6, - 5 and 8 would be a coefficient.0055

We often like to organize things from the highest power to the smallest power.0068

The highest power here, it is a special name we will call this the leading coefficient.0073

This will be important terms you will often here me use later on when talking about polynomials.0083

Recall that things are like terms if they have the exact combination of variables with the same exponents.0092

In this giant list here, I have lots and lots of examples of like terms.0100

This first one is an example of like terms because they both have an m3 and notice it has no difference what the coefficient out front is.0106

I have 19, 14 but it does not matter that part.0117

What does matter is that I they both have an m3.0121

The next two have both y9 and if you have more than one variable in there then both of those variables better matchup.0127

Both of them have a single x and they both have a y2.0136

If we understand polynomials and understand terms then you can start understanding polynomials.0144

A polynomial is a term or a finite sum of terms in which all the variables have whole number exponents and no variables up here in the denominator.0150

To make this a little bit more clear, I have many different examples of polynomials and many different examples of things that are not polynomials.0160

I will pick these over and make sure that they fit the definition.0168

This first one here, I know that it is a polynomial as I can see that has a finite number of terms that means that stops eventually.0174

If I look at all the exponents present like the two, then all of those are nice whole numbers and I do not see anything in the denominator.0182

In fact, there are no fractions there and we do not have any variables in the denominator.0191

That is why that one is a polynomial.0196

I will put in this next example to highlight that you could have coefficient that are fractions0201

but the important part is that we do not have variables in the denominator, that would make it not a polynomial.0206

It is okay if we have more than one variable like start mixing around m and p, just as long as the exponents on those state nice whole numbers.0216

Even the next one is a good example of this.0227

I have y and x, but I have a finite number and eventually stops and it looks like I do not have any variables in the bottom.0229

Now a lot of things get to be a polynomial, even very small things.0239

For example, something like 4x is a type of polynomial, it is not a very big polynomial and has exactly one term, but is finite.0245

All of the exponents are nice whole powers and there are no variables in the denominator.0253

You can have even very, very short polynomials.0259

This one is just the number 5 has no variables, or even consider it x0.0261

Compare these ones to things that are not polynomials and watch how they break the definition in some sort of way.0267

In this first one is not very big, it is 1 ÷ x and it is not polynomial because we are dividing by x.0276

We do not want that x in the bottom.0282

This next one is not a polynomial because of its exponents.0286

It has an exponent of ½ and another exponent of 1/3 and because of those exponents, it is not a polynomial.0290

The next one is a little tricky.0299

It looks like it should be a polynomial, I mean I have 1, 2x to the first and 3x2.0301

The reason why this is not a polynomial has to do with this little…0307

That indicates that this keeps going on and on forever.0311

In order to be a polynomial, it should stop.0315

It should be finite somewhere.0318

That one is not a polynomial.0319

Other things that we want to watch out for is we do not have any of our variables and roots and none of those exponents should be negative.0323

That should give you a better idea of when something is a polynomial or not a polynomial.0334

We can start to classify the types of polynomials we have by looking at two aspects.0342

One is how many terms they have.0348

If it only has one term we would call that a monomial.0351

In example 3x, it only has a single term, it is a monomial.0357

If it has two terms then we can call that one a binomial, think of like a bicycle or something like that, two terms.0363

If I have 3 terms we will go ahead and call it a trinomial.0374

Usually if it has four or more terms you will get a little bit lazy we just usually call those as polynomials.0384

Technically, all of these are examples of polynomials but they are just a very specific type of polynomial.0391

Let me write that one with a bunch of different terms, 5x4 - 3x3 + x2 - x +7 that would be a good example of a polynomial.0398

In addition to talking about the monomial and binomial, that fun stuff, you can also talk about the degree of a polynomial.0413

What the degree is, it is the highest power of any nonzero terms.0421

You are looking for that biggest exponent.0426

In this first one here, you can see that the largest exponent is 2, we would say that this is a 2nd degree polynomial.0429

In the next one, just off to the right, the largest power in there is a 3, so this would be a 3rd degree polynomial.0440

Now you can combine these two schemes together and get specific on the types of polynomials you are talking about.0452

Not only for that first one can I say it is a 2nd degree because it has the largest power of 2 in there,0459

but I can say it is a 2nd degree binomial because it has two terms in it.0465

With the other one in addition to saying it is a 3rd degree polynomial, I can take a little further and say it is a 3rd degree trinomial.0473

That is because we have 1, 2, 3 terms present in a polynomial.0484

To evaluate a polynomial it is a lot like evaluating functions.0494

We simply take the value that we are given and we substitute it for all copies of the variables.0498

Let us give that a try with 2y3 + 8y - 6 and we want to evaluate it for y = -1.0504

I will go through and everywhere I see a copy of y, we will put in that -1.0513

Let us go ahead and work on simplifying this.0527

-13 would be -1 × -1× -1,that would simply be -1.0532

8 × -1 would be -8 and then we have -6.0539

Continuing on, I have -2 – 8 – 6 = -16.0546

If you are evaluating it, just take that value and substitute it in for all copies of that variable.0557

Onto what we wanted to, adding and subtracting polynomials.0568

When we get into addition and subtraction, what we are looking to do is add or subtract the like terms present in the polynomial.0573

There are two ways you should go about this and both of them are perfectly valid so you will use whichever method you are more comfortable with.0581

Here I want to add the following two polynomials.0590

The way I’m going to do this is I'm going to simply highlight which terms are like terms.0593

Here is 4x3 and 6x3 those are like terms - 3x2 2x2, those are like.0599

2x and - 3x and so one by one we will take these like terms and simply put them together.0610

4x3 and 6x3 = 10x3.0619

-3x2 and 2x2 would be -1x2.0626

2x - 3x =- 1x.0633

You can see I have all of the parts there and it looks like I am left with a 3rd degree trinomial.0640

The other way that you can do this, if you are a little bit more comfortable with it, is you can take one polynomial over the other polynomial.0645

If you do it this way, you want to make sure you line up what are your like terms.0660

Put your x3 on your x3, your x2 on your x2 and you x on your x.0665

It is essentially the same idea and that you go through adding all of the terms as you go along.0672

I will start over on the right side of this one, 2x and I'm adding - 3x, there is - x for that one.0678

Onto the next set of terms, - 3x2 + 2x2 = - x2 and 4x3 and 6x3 = 10x3.0688

You can see you get exactly the same answer but just use whatever method you are more familiar with.0701

Let us try some examples.0711

In this one we want to determine if these are polynomials or not.0712

Let us try that first one.0718

I see I have 2x + 3x2 – 8x3.0719

The things we are watching out for is, one does it stop.0723

I do not see any… out here, I know this is finite, that looks pretty good.0727

I do not see any variables in the denominator, so that is good.0732

There are no fractions with the x's in the bottom.0736

All of the exponents here, the 1, 2, and the 3 all of those are nice whole numbers.0740

This one is looking good.0747

I will say that this one is a polynomial.0749

2/x + 5/4x2 – x3/6.0757

In this one I can think I can see a problem right away.0762

Notice how we have variables in the bottom and because of that I will say that this one is not a polynomial.0765

Be careful and watch out for that criteria.0776

This next example, we want to just go through and classify what types of polynomials these are.0782

We use two criteria for this, we look at its degree and we will see how many terms it has.0788

The first one, the largest power I can see in here is this 3.0794

I will say this is a 3rd degree.0798

It is a polynomial but let us be a little more specific.0811

It has 1, 2, 3 terms, so I will say this is a 3rd degree trinomial.0814

Let us try another one, the largest power here is 4, 4th degree.0826

It only has 1, 2 terms, so it will be our binomial.0838

One more to classify, this one has a bunch of different exponents, but of course the largest one is the only one we are interested in.0847

This is a 3rd degree and now we count up all of its terms, 1, 2, 3, 4.0856

Since it has four terms, I will just keep calling this one a polynomial.0870

Let us get into adding the following polynomials together.0883

The way I’m going to do this is I’m going to line them up, one on top of the other.0886

Starting with the first one, I have 3x3, I do not have any x2, I have 4x and I have 1.0892

All of that would represent my first polynomial there.0905

Below that I want to write the second polynomial but I want to lineup the terms, -3x2 I will put it under the other x2.0910

I have a 6x I will put that under the other x and the 6 I will go ahead and put that with the 1.0922

You can now have things all nice and lined up.0931

Let us go ahead and add them one at a time.0934

1 + 6 = 7, -4 + 6x =2x, 0x2 + -3x2 = -3x2.0938

One more, this has nothing to add to it so just 3x2.0961

This would be our completed polynomial after adding the two together.0967

One last example is we are going to work on subtracting polynomials.0977

With these ones, what I suggest is being very careful with your signs.0981

You will see that I’m going to start with lining one on top of the other one.0985

But I’m going to end up distributing my negative signs, I will turn this into an addition problem.0989

Let us give it a try.0995

The polynomial on the left is 7y2 -11y + 8 and right below that is -3y2 + 4y + 6.0996

Now comes the important part, we are subtracting these polynomials so I will put a giant minus sign up front.1015

Before getting too far, I could go through and try and subtract this term by term,1023

but it is much easier to distribute my negative sign and just look at this like an addition problem.1028

My top polynomial will stay unchanged and we will leave that as it is.1036

After distributing the bottom, here is what we get negative × - 3y = 3y2, negative × 4y = -4y and for the very last one -6.1043

We can take care of this as an addition problem and we know that the subtraction is taken care of because we put it into all of our terms.1060

8 + -6 = 2, -11y + -4y = -15y and then I have a 7y2 + 3y2 = 10y2.1071

This final polynomial represents the two being subtracted.1098

Now you know a little bit more about polynomials and have put them together.1105

Remember that you are just combining your like terms.1109

Thank you for watching www.educator.com.1112

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