Polynomial Operations

Algebra 1 – Versal Course http://bit.ly/2dYzTd0 Polynomial Operations
Notes and problems adapted from Trapper Hallam and Prentice Hall Mathematics Algebra 1 text, 2004.
Polynomial Operations Unit
Suggested Pacing & In-Class Activities
Date In-Class At Home
2/1
 An Introduction to Blended Learning
 Getting Started
 Adding & Subtracting Polynomials Notes
 Distributive Property Notes
2/2
 Adding & Subtracting Polynomials Digital
Assignment
 Distributive Property Digital Assignment
 Degree & Standard Form Notes
 Degree & Standard Form Digital
Assignment
2/3
 Naming Polynomials Notes
 Naming Polynomials Digital Assignment
 Catch up
2/6
 Quiz
 Greatest Common Factor (GCF) Notes
 GCF Digital Assignment
2/7  ACTIVITY: GCF Bingo
 GCF Factoring Notes
 GCF Factoring Mastery Activity (Part 1)
2/8
 ACTIVITY: GCF Factoring Mastery Activity
(Part 2)
 GCF Factoring Digital Assignment
 GCF Factoring Mastery Activity (Part 3)
 Multiplying Polynomials Notes
2/9
 Multiplying Polynomials Digital
Assignment
 Floor Plan Project Intro
 Review
2/10  Floor Plan Project
 Review
 Floor Plan Project
2/13  Floor Plan Project
 Review
2/14  Floor Plan Project  Floor Plan Project
2/15
 Quiz
 Finish Floor Plan Project

Polynomial Operations Unit Checklist
Adding & Subtracting Polynomials
Notes (20 minutes)
Practice Problems
Digital Assignment
Distributive Property
Notes (5 minutes)
Practice Problems
Digital Assignment
Degree & Standard Form
Notes (20 minutes)
Practice Problems
Digital Assignment
Naming Polynomials
Notes (15 minutes)
Practice Problems
Digital Assignment
Monday Quiz
Greatest Common Factor
Notes (25 minutes)
Practice Problems
Digital Assignment
Tuesday GCF Bingo
GCF Factoring
Notes (25 minutes)
Mastery Activity (Part 1 – Before Class)
Mastery Activity (Part 2 – During Class) Wednesday 2/8
Practice Problems
Digital Assignment
Mastery Activity (Part 3 – After Class)
Multiplying Polynomials
Notes (20 – 30 minutes)
Practice Problems
Digital Assignment
Review
Floor Plan Project (3 days in class)
Quiz Wednesday 2/15

Notes – Adding & Subtracting Polynomials Name:
1Standard: Date: Hour:
Definitions
Base:
Coefficient:
Exponent:
Like Terms:
Adding Polynomials
1. (12𝑚2
+ 4) + (8𝑚2
+ 5) 2. (4𝑥2
+ 𝑥 + 7) + (2𝑥2
− 6𝑥 + 1)
3. 2𝑗3
+ 𝑗2
+ 4
+ (𝑗3
− 3𝑗2
+ 7𝑗 − 15)
Subtracting Polynomials
4. (10𝑦2
+ 6) − (4𝑦2
+ 5) 5. (6𝑣2
− 𝑣) − (7𝑣2
− 3𝑣)
6. 2𝑤3
+ 5𝑤2
− 3𝑤
− (𝑤3
− 8𝑤2
+ 11𝑤)

Practice – Adding & Subtracting Polynomials Name:
Practice
Simplify each expression by adding or subtracting the polynomials.
1. 5𝑚2
+ 9
+ 3𝑚2
+ 6
2. 𝑤2
+ 𝑤 − 4
+ 7𝑤2
− 4𝑤 + 8
3. (8𝑥2
+ 1) + (12𝑥2
+ 6) 4. (𝑎2
+ 𝑎 + 1) + (5𝑎2
− 8𝑎 + 20)
5. 6𝑐 − 5
− (4𝑐 + 9)
6. 7ℎ2
+ 4ℎ − 8
− (3ℎ2
− 2ℎ + 10)
7. (17𝑛4
+ 2𝑛3) − (10𝑛4
+ 𝑛3) 8. (6𝑤2
− 3𝑤 + 1) − (𝑤2
+ 𝑤 − 9)
9. (−5𝑥4
+ 𝑥2) − (𝑥3
+ 8𝑥2
− 𝑥) 10. (7𝑦3
− 3𝑦2
+ 4𝑦) + (8𝑦4
+ 3𝑦2)
11.
3𝑘 − 8 7𝑘 + 12
What is the total area of the two rectangles?
12. Yard A has a perimeter of 2𝑏 + 6 feet. Yard B
has a perimeter of 𝑏 + 5 feet. How much
larger is Yard A’s perimeter than Yard B’s?

Notes – Distributive Property Name:
Distributive Property
For every real number a, b, and c,
Examples
Simplify each expression by using the distributive property.
1. 3(2𝑥 + 5) 2. 4𝑥(3𝑥 + 2)
3. (𝑥 + 7)4 4. 2𝑥(4𝑥 − 9)

Practice – Distributive Property Name:
Simplify each expression.
1. 8(3𝑘 + 6) 2. (𝑥 − 5)9
3. 2𝑑(𝑑 + 1) 4. −6(7𝑥 + 4)
5. −7𝑤(3𝑤 − 8) 6. 4𝑦(5𝑦2
+ 𝑦 + 6)
7. Find the area of the figure.
3𝑥2
2𝑥−7

Notes – Degree & Standard Form Name:
Which Polynomial is “bigger” – A or B? Why?
Definition
A ___________________is an expression that is a number, a variable, or a product of a number and one or
more variables.
The ___________________of a monomial is the sum of the exponents of its variables. For a nonzero
number, the degree is 0. Zero has no degree.
Examples:
1. Find the degree of each monomial.
a.
2
3
𝑥8 b. 7𝑥2
𝑦3
c. −4 d. 6𝑐 e. 3𝑥𝑦3
Definition
A polynomial is a monomial or the sum or difference of two or more monomials.
______________________ ______________of a polynomial means that the degrees of its monomial terms
decrease from left to right.
Standard Form
1. Combine like terms
2. Degree decreases from left to right
Examples
2. Write each polynomial in standard form.
a. 5 − 2𝑥 b. 3𝑥4
− 4 + 2𝑥2
+ 5𝑥4
c. −2 + 7𝑥 d. 3𝑥5
− 202𝑥5
+ 7𝑥

Practice – Degree & Standard Form Name:
Find the degree of each monomial
1. 4𝑥 2. 7𝑐3
3. −16
4. 6𝑦2
𝑤8
5. 6 6. 11𝑘
Write each polynomial in standard form
7. 4𝑥 − 3𝑥2
8. 4𝑥 + 9
9. 𝑐2
− 2 + 4𝑐 10. 9𝑧2
+ 5𝑧 − 11𝑧2
− 5
Add or subtract the polynomials. Write your final answer in standard form.
11. (7𝑦2
− 3𝑦 + 4𝑦) + (8𝑦2
+ 3𝑦2
+ 4𝑦) 12. (2𝑥3
− 5𝑥2
− 1) − (8𝑥3
+ 3 − 8𝑥2)
13. (−7𝑧3
+ 3𝑧 − 1) − (−6𝑧2
+ 𝑧 + 4) 14. (7𝑎3
− 𝑎 + 3𝑎2) + (8𝑎2
− 3𝑎 − 4)
15. Find the perimeter of the figure. Write your answer in standard form.
Remember, perimeter is the total distance around the shape.

Notes – Naming Polynomials Name:
Definition
The degree of a polynomial in one variable is the same as the degree of the monomial with the greatest
exponent. **Note** When a polynomial is written in standard form, the degree of the first term is the
degree of the polynomial.
Examples
Polynomial Degree
3𝑥2
+ 4𝑥 − 5 2
5𝑥7
+ 13𝑥5
− 𝑥2
+ 6 7
Naming Polynomials
Degree Name by Degree Polynomial Example(s)
# of Terms Name by Terms Polynomial Example(s)
1. Name each expression based on its degree and number of terms.
a. −2𝑥 + 5 b. 8𝑥4
+ 2𝑥2
− 4 c. 7𝑥 − 2 d. −199𝑥5
+ 7𝑥

Practice – Naming Polynomials Name:
Name each expression based on its degree and number of terms.
1. 5𝑥2
− 2𝑥 + 3 2.
3
4
𝑧 + 5 3. 7𝑎3
+ 4𝑎 − 12
4. 6𝑥7
− 4𝑥3
− 11𝑥 + 5 5. −15 6. 𝑤2
+ 2
Write each expression in standard form, then name each polynomial based on its degree and number of
terms.
7. 4𝑥 − 3𝑥2
8. 4𝑥 + 9 9. 𝑐2
− 2 + 4𝑐
10. 9𝑧2
− 11𝑧2
+ 5𝑧 − 5 11. 𝑦 − 7𝑦3
+ 15𝑦8
12. −10 + 4𝑞4
− 8𝑞 + 3𝑞2
Simplify. Write each answer in standard form, then name the polynomial based on its degree and number of
terms.
13. (𝑥3
+ 3𝑥) + (12𝑥 − 𝑥4) 14. (6𝑔 − 7𝑔8) − (4𝑔 + 2𝑔3
+ 11𝑔2)

Notes – Greatest Common Factor (GCF) Name:
Definition
To factor a number means to break it up into natural numbers that can be _________________ together to
get the original number. Factoring is like undoing multiplication.
Example
1. Find all the factors for the given number.
a. 4 b. 18 c. 24
Definition
The greatest common factor (GCF) of two or more numbers is the largest number that divides each of the
numbers.
Example
2. Find the greatest common factor of the following pairs of numbers
a. 8, 12 b. 16, 30 c. -50, 60
We can also find the GCF of monomials by factoring the numbers and variables that make up the monomial.
3. Find the GCF of the monomials
a. 𝑥2
, 𝑥6
b. 4𝑦2
, 2𝑦3
c. −16𝑡5
, 30𝑡2
The GCF of a polynomial is the GCF of the individual terms
4. Find the GCF of the polynomial.
a. 5𝑐5
+ 10𝑐3
b. 4𝑎3
− 2𝑎2
− 6𝑎

Practice – Greatest Common Factor (GCF) Name:
Find the GCF for each of the following problems.
1. 39, 6 2. 24, 29
3. 40, 10 4. 39𝑣, 30
5. 39𝑛2
, 21𝑛 6. 30𝑦3
, 20𝑦2
7. 8𝑥2
+ 10𝑥 8. 12𝑦 − 16
9. −15𝑑5
+ 45𝑑3
10. 𝑐3
+ 𝑐2
− 𝑐
11. 6𝑛2
− 30𝑛 + 42 12. 18𝑝3
− 63𝑝2
− 9𝑝

Activity – GCF Bingo Name:
B I N G O
FREE
SPACE

Notes – GCF Factoring Name:
1. Simplify each product
a. 4(5𝑏2
+ 𝑏 + 6) b. 𝑡(𝑡 + 3)
c. 4𝑦2(5𝑦4
− 3𝑦2
+ 2) d. −3𝑣3(6𝑣4
+ 4𝑣2
− 𝑣 − 5)
Note
To factor a polynomial:
- Find the GCF
- Write the GCF multiplied by the remaining factors of each term of the polynomial.
 i.e. divide the GCF from each of the terms
- Check Twice
 Is there a GCF remaining?
 Check your answer using the distributive property
2. Factor each polynomial. Check your answer.
a. 3𝑡2
− 18 b. 5𝑣5
+ 10𝑣3
c. 4𝑏3
+ 2𝑏2
+ 6𝑏 d. 6𝑚3
− 12𝑚2
+ 24𝑚

Practice – GCF Factoring Name:
Factor the GCF from each of the polynomials. Check your answers using the distributive property.
1. 6𝑥 − 4 2. 𝑣2
+ 4𝑣
3. 15𝑛3
− 3𝑛2
+ 12𝑛 4. 6𝑝6
+ 24𝑝5
+ 18𝑝3
5. 7𝑥2
+ 6𝑥 − 3 6. 15𝑥6
− 30𝑥5
− 75𝑥3
+ 90𝑥2
7. Find the dimensions of the rectangle with the given area.
10𝑥3
− 25𝑥2
+ 20
8. Katie does the following work, circle the error and explain what she did incorrectly. Then correctly
factor the GCF from the polynomial
5𝑥4
+ 4𝑥3
+ 3𝑥2
𝑥2(5𝑥2
+ 4𝑥3
+ 3𝑥2)
9. Steven factors the GCF from the following polynomial. Circle and explain his error, then factor the GCF
from the polynomial correctly.
24𝑛3
− 96𝑛2
+ 48𝑛
8𝑛(16𝑛2
− 88𝑛 + 40)

Notes – 9.3 Multiplying Polynomials Name:
There are 3 ways that you can multiply polynomials: the distributive property, the box method, or the vertical
method. Videos for each method are online. After you have watched the multiplying polynomials intro video,
decide which method you’d like to use and watch that specific video. You might choose to practice 2
methods.
Examples
1. (2𝑥 − 3)(𝑥 + 5) 2. (𝑎 + 7)(𝑎 + 4)
3. (3𝑥 − 4)(𝑥 + 2) 4. (2𝑦 − 5)(3𝑦 + 7)
5. (2𝑘 − 5)2
6. (4𝑥2
+ 𝑥 − 6)(2𝑥 − 3)

Practice – Multiplying Polynomials Name:
Simplify each product using any method.
1. (𝑎 + 5)(𝑎 − 2) 2. (3𝑥 − 4)(2𝑥 − 4)
3. (2𝑥2
+ 𝑥)(5𝑥 + 7) 4. (3𝑥2
− 6𝑥 − 2)(2𝑥 + 1)
5. (𝑥 + 3)(𝑥 − 3) 6. (3𝑥 + 5)2

Find the area of each figure
7. 8.
9. Assume this is a square 10. Find the area of the shaded region. Write your
answer in standard form.
Hint: how much more area does the large box
have than the small box?
𝑦 − 5
3𝑦 + 2
𝑘 + 2
3𝑘 + 19
𝑗 + 8
𝑥 + 2
𝑥 + 3
𝑥 − 3
𝑥

Review – Polynomial Operations Name:
Write the polynomial in Standard Form. Then name the polynomial based on its degree and number of terms.
1. 2 − 11𝑥2
− 8𝑥 + 6𝑥2
2. 3𝑥 − 4𝑥2
+ 𝑥3
+ 𝑥
3. 5 − 2𝑥 + 𝑥2
4. 6 − 2𝑥 + 4 − 3𝑥
5. 7𝑥4
− 5𝑥5
+ 2𝑥4
+ 8𝑥5
Simplify
6. (−7𝑥 − 5𝑥4
+ 5) + (−7𝑥4
− 5 − 9𝑥) 7. (3𝑥3
− 2𝑥2
− 𝑥) + (4𝑥4
− 3𝑥2
+ 5)
8. Write the perimeter of the figure
9. (−7𝑥 − 5𝑥4
+ 5) − (−7𝑥4
− 5 − 9𝑥) 10. (3𝑥2
+ 𝑥3
− 𝑥 + 4) − (3𝑥3
+ 𝑥2
− 2𝑥 + 5)

11. 6(𝑥3
− 2𝑥2
+ 4𝑥) 12. 𝑥2(2𝑥3
− 3)
13. 3𝑥(4𝑥2
− 5𝑥 + 8) 14. 2𝑥3(5𝑥4
− 2𝑥 − 6)
Factor the GCF
15. 3𝑥3
− 15𝑥2
+ 18𝑥 16. 2𝑥 − 6
17. 24𝑤12
+ 64𝑤8
18. 54𝑦7
+ 9𝑦6
Multiply
19. (𝑥 − 4)(𝑥 + 2) 20. (3𝑥 − 4)(2𝑥 + 6)
21. (3𝑥 − 7)(3𝑥 − 5) 22. (4𝑥 + 3)(2𝑥 + 5)

23. (5ℎ − 6)(5ℎ − 6) 24. (𝑥 + 2)(3𝑥 − 5)
25. (2𝑛2
+ 4𝑛 + 4)(4𝑛 − 5) 26. (𝑘 − 5)(3𝑘2
+ 4𝑘 − 7)
27. (𝑥 − 1)2
28. (2𝑤 − 5)2
29. (3𝑛 + 4)2
30. (𝑥 + 7)(𝑥 − 7)

31. (2𝑦 − 3)(2𝑦 + 3) 32. (4𝑚2
− 5)(4𝑚2
+ 5)
33. Find the area of the unshaded region. Write your answer in standard form.
34. The Johnsons want to cover their backyard with new grass. Their backyard is rectangular, with a
length of 3𝑥 − 5 and a width of 4𝑥 − 10 feet. However, their rectangular swimming pool and its
surrounding patio has dimensions of 𝑥 + 8 by 𝑥 − 2 feet. What is the area of the region of the yard
that they want to cover with new grass?
Hint: draw a diagram, then label it.
x + 5
x

Floor Plan Project Name:
This project will explore floor plans. You will be creating a floor plan of the classroom. We will have 3 days in
class to work on the project plus any remaining time after your quiz. The project will be a summative
assessment.
Here are a couple videos that explain a bit about floor plans. These videos are done by John Brown of Slow
Home Studio, a Canadian company that focuses on design education for homeowners.
Part 1 - https://youtu.be/YwqI7zyS-4A Part 2 - https://youtu.be/Lh6HSkAxd8c
Rather than conventional measurements, our floor plans will incorporate polynomials as measurements. You
will need any item you’d like to use, as long as you can reasonable measure the item from one end to the
other. It should be larger than a penny but smaller than a standard sheet of paper. You will use this item as
your measuring device. I will demonstrate how in class.
Supplies
 Ruler (will be provided for in class use)
 An object (driver’s license, school ID, water bottle, phone, anything)
 Floor Plan Starter sheet (provided)
 Project turn in sheet (provided)
Guidelines for Success
 Determine the dimensions of each required item (length & width; classroom dimensions will be given)
 Find the area of each item (A=l*w)
 Find the total area of all the furniture in the classroom
 Find the amount of empty floor space
 Draw up your classroom floor plan and include:
o 30 student desks o 1 teacher desk o 1 filing cabinet o 3 tables
 Label each item: abbreviations, color codes, or shapes may be helpful here, remember to include a key
 Label the dimensions of one of each item (you do not need to put dimensions for each student desk)
Creativity
You may be creative with the floor plan. You do not need to set up the classroom as it is normally – you may
arrange the desks in groups, rows, pairs, etc., or move the tables, file cabinet, and Ms. Heizer’s desk to
another location in the classroom. Make sure all the required items are included in your floor plan. If you’d
like, you may color your floor plan to enhance its visual appearance, but make sure all the labels are easy to
see and read.
Neatness will contribute to your grade. This means use a straight edge, I do not want to see jagged lines. If
you chose to color your floor plan, neatness will count here too. As always, showing your work will count
towards your grade. Do not try to cheat off a friend – each of you are using different items, so no two
students should have the same dimensions, areas, or answers.
If you would prefer to do this assignment digitally, please see Ms. Heizer to discuss the options – this is NOT
for every student.

Rubric – Floor Plan Project Name:
Item used:_________________________ Length of its edge: _____________
All Items Included in Floor Plan
7⁄Student Desks (30)
Teacher’s Desk (1)
File Cabinet (1)
Tables (3)
Items Labeled on Floor Plan
5⁄Student Desk (30)
Teacher’s Desk (1)
File Cabinet (1)
Table (3)
Key or Abbreviations Used
Dimensions Labeled on Floor Plan
8⁄Student Desk
Teacher’s Desk
File Cabinet
Table
Project Turn-in Sheet
25⁄
Neatness
5⁄
Total
50⁄

Floor Plan Project – Turn-in Sheet Name:
Student Desk Work
Length:
Width:
Area:
Teacher’s Desk Work
Length:
Width:
Area:
File Cabinet Work
Length:
Width:
Area:
Table Work
Length:
Width:
Area:
Classroom Work
Length: Width:
Area:

Total area of all furniture in classroom:
Amount of empty floor space in classroom:

Name: _______________________________________ Hour: __________
25’
40’

Polynomial Operations

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