Section 6: Polynomials

Foundations of Math 9 Updated June 2019

Adrian Herlaar, School District 61 www.mrherlaar.weebly.com

Section 6: Polynomials

This book belongs to: Block:

Section Due Date Date Handed In Level of Completion Corrections Made and Understood

𝟔. 𝟏

𝟔. 𝟐

𝟔. 𝟑

Self-Assessment Rubric

Learning Targets and Self-Evaluation

L – T Description Mark

𝟔 − 𝟏 Understanding terms, degree, coefficients, and constants

Grouping like terms

Pictorially demonstrating terms using algebra tiles

𝟔 − 𝟐 Applying integer fundamentals in addition and subtraction of polynomials

Applying exponent laws in the multiplication and division of polynomials

Performing combined Operations of polynomials

Comments:

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

Category Sub-Category Description

Expert (Extending)

4 Work meets the objectives; is clear, error free, and demonstrates a mastery of the Learning Targets

“You could teach this!”

3.5 Work meets the objectives; is clear, with some minor errors, and demonstrates a clear understanding of the Learning Targets

“Almost Perfect, one little error.”

Apprentice (Proficient)

3 Work almost meets the objectives; contains errors, and demonstrates sound reasoning and thought

concerning the Learning Targets

“Good understanding with a few errors.”

Apprentice (Developing)

2 Work is in progress; contains errors, and demonstrates a partial understanding of the

Learning Targets

“You are on the right track, but key concepts

are missing.”

Novice (Emerging)

1.5 Work does not meet the objectives; frequent errors, and minimal understanding of the Learning

Targets is demonstrated

“You have achieved the bare minimum to meet the learning outcome.”

1 Work does not meet the objectives; there is no or minimal effort, and no understanding of the

Learning Targets

“Learning Outcomes not met at this time.”

Foundations of Math 9

1 Adrian Herlaar, School District 61 www.mrherlaar.weebly.com

Competency Evaluation

A valuable aspect to the learning process involves self-reflection and efficacy. Research has shown that authentic

self-reflection helps improve performance and effort, and can have a direct impact on the growth mindset of the

individual. In order to grow and be a life-long learner we need to develop the capacity to monitor, evaluate, and

know what and where we need to focus on improvement. Read the following list of Core Competency Outcomes

and reflect on your behaviour, attitude, effort, and actions throughout this unit.

4 3 2 1

I listen during instruction and come ready to ask questions

Personal Responsibility

I am on time for class

I am fully prepared for the class, with all the required supplies

I am fully prepared for Tests

I follow instructions keep my Workbook organized and tidy

I am on task during work blocks

I complete assignments on time

I keep track of my Learning Targets

Self-Regulation

I take ownership over my goals, learning, and behaviour

I can solve problems myself and know when to ask for help

I can persevere in challenging tasks

I am actively engaged in lessons and discussions

I only use my phone for school tasks

Classroom

Responsibility and

Communication

I am focused on the discussion and lessons

I ask questions during the lesson and class

I give my best effort and encourage others to work well

I am polite and communicate questions and concerns with my peers and teacher in a timely manner

I clean up after myself and leave the classroom tidy when I leave

Collaborative Actions

I can work with others to achieve a common goal

I make contributions to my group

I am kind to others, can work collaboratively and build relationships with my peers

I can identify when others need support and provide it

Communication

Skills

I present informative clearly, in an organized way

I ask and respond to simple direct questions

I am an active listener, I support and encourage the speaker

I recognize that there are different points of view and can disagree respectfully

I do not interrupt or speak over others

Overall

Goal for next Unit – refer to the above criteria. Please select (underline/highlight) two areas you want to focus on

Rank yourself on the left of each column: 4 (Excellent), 3 (Good), 2 (Satisfactory), 1 (Needs Improvement)

I will rank your Competency Evaluation on the right half of each column



Section 6.1 – Polynomials

Vocabulary

There will be a lot vocabulary necessary to accurately understand Polynomials

Term: Any variable, constant, or product of the two

Example: 3, 4𝑥, 𝑡, 2𝑟2, 𝑥𝑦𝑧

Like Terms: Terms that have the same variable(s) to the same exponents

Example: 𝑥2 𝑎𝑛𝑑 4𝑥2, 7𝑡 𝑎𝑛𝑑 3𝑡, 4 𝑎𝑛𝑑 9

Degree of a Term: The exponent on the variable or sum of exponents on different variables

of one term

Example: 3𝑥 𝑖𝑠 𝐷𝑒𝑔𝑟𝑒𝑒 1, 4𝑥2 𝑖𝑠 𝐷𝑒𝑔𝑟𝑒𝑒 2, 5𝑥𝑦𝑧 𝑖𝑠 𝐷𝑒𝑔𝑟𝑒𝑒 3

Polynomial: Any term or terms separated by addition or subtraction where all

exponents on the variables are whole numbers

Example: 5𝑡2 + 2𝑡 − 7

Leading Term: The term in a Polynomial with the highest degree

Example: From above: 5𝑡2 is the leading term, it has the highest degree

Descending Order: Writing terms from highest to lowest degree

Example: From Above: Is in descending order, degree goes 2, 1, 0

Polynomial Degree: The highest degree on a term, becomes the degree of the polynomial

Example: From Above: 5𝑡2 is the leading term with degree 2, so the

Polynomial is of degree 2



Combining like Terms

Combining like terms is doing exactly that

When we have a long list of terms written as a Polynomial, we can combine any that are

Like Terms: same variables, same exponent

Example: 3𝑞2 − 4 + 2𝑞 − 2𝑞2 + 4𝑞 − 8

Like Terms are: 3𝑞2 𝑎𝑛𝑑 −2𝑞2 , − 4 𝑎𝑛𝑑 − 8, 2𝑞 𝑎𝑛𝑑 4𝑞

So, 3𝑞2 − 2𝑞2 = 𝑞2

2𝑞 + 4𝑞 = 6𝑞

−4 − 8 = −12

And Descending Order: 𝒒𝟐 + 𝟔𝒒 − 𝟏𝟐

Example: Combine the Like Terms and leave the simplified expression in Descending Order

5𝑥𝑦 + 5𝑥2 + 2𝑥 − 6 − 4𝑦𝑥 + 2𝑥 + 6 − 3𝑥2

Like Terms are:

+5𝑥2𝑎𝑛𝑑 −3𝑥2 so 5𝑥2 − 3𝑥2 = 2𝑥2 𝐷𝑒𝑔𝑟𝑒𝑒 𝑜𝑓 2

5𝑥𝑦 𝑎𝑛𝑑 − 4𝑦𝑥 so 5𝑥𝑦 − 4𝑥𝑦 = 𝑥𝑦 𝐷𝑒𝑔𝑟𝑒𝑒 𝑜𝑓 2

+2𝑥 𝑎𝑛𝑑 + 2𝑥 so 2𝑥 + 2𝑥 = 4𝑥 𝐷𝑒𝑔𝑟𝑒𝑒 𝑜𝑓 1

−6 𝑎𝑛𝑑 + 6 so −6 + 6 = 0 𝐷𝑒𝑔𝑟𝑒𝑒 𝑜𝑓 0

Since 𝑥2𝑎𝑛𝑑 𝑥𝑦 are both degree 2, which one goes first?

We list them ALPHABETICALLY, 𝑥2 = 𝑥𝑥 𝑎𝑛𝑑 𝑥𝑥 𝑐𝑜𝑚𝑒𝑠 𝑏𝑒𝑓𝑜𝑟𝑒 𝑥𝑦

𝟐𝒙𝟐 + 𝒙𝒚 + 𝟒𝒙

Degree 2 Degree 1

Degree 0

𝑥𝑦 and 𝑦𝑥 are the same, in

multiplication order doesn’t matter,

𝑥𝑦 = 𝑦𝑥



Section 6.1 – Practice Questions

Identify the number of terms, what are they, and their degrees?

1. 3𝑥 − 4𝑥2 − 5

2. 4𝑥𝑦𝑧

3. −2𝑥𝑦𝑧 − 5𝑥𝑦 + 4

4. 5𝑥3𝑦 + 4𝑥𝑦3 − 6𝑥𝑦𝑧 5. 5

6. 3𝑥 + 4𝑦 + 5𝑧 − 𝑥2

Put the following Polynomials in DESCENDING ORDER

7. 3 + 4𝑥2 − 5𝑥 8. −2𝑡 + 4𝑡3 − 2𝑡 − 3𝑡2

9. 2 − 𝑥 + 5𝑥2

10. 𝑧2 − 4𝑧 + 5 11. 𝑥 + 𝑥𝑦 + 𝑥𝑧 − 𝑦

12. −5𝑥𝑦 − 𝑦 + 2𝑥 + 𝑥2



Simplify the following, put your answer in DESCENDING ORDER

13. 3𝑡 + 4 − 6𝑡 + 2𝑡2 + 4 − 3𝑡2 14. 7𝑧3 + 2𝑧 − 4𝑧2 + 1 − 4𝑧2 − 5 + 3𝑧2 + 3𝑧 15. 5𝑥𝑦 + 3 − 5𝑦𝑥 + 2 16. −4𝑞 + 5𝑞2 − 7 − 5𝑞2 + 4𝑞 + 7



17. 1

3𝑖2 + 2𝑖 −

1

6𝑖2 + 4 − 9

18. −4.9𝑥 − 3.2𝑦 − 1.3𝑥 + 4.2𝑦 + 1

19. 11

5𝑥 +

2

3𝑦 −

3

5𝑥 −

1

3𝑦 + 10

20. 1

4𝑗2 − 𝑗 −

1

2𝑗 +

3

8𝑗2 +

5

16𝑗2



Section 6.2 – Addition and Subtraction of Polynomials

All we are doing here is grouping like terms, however it involves a couple extra steps

Let’s start with Addition

Addition of Polynomials

Example: (𝑥2 + 4𝑥 − 7) + (2𝑥2 − 3𝑥 + 4)

We have 2 Polynomials, shown in brackets, and we are adding the second Polynomial to the first.

Step 1: In addition just drop the Brackets, keep the sign on the first term of the second

Polynomial, since it’s positive, nothing changes

𝑥2 + 4𝑥 − 7 + 2𝑥2 − 3𝑥 + 4

Step 2: Group the Like Terms

𝑥2 + 4𝑥 − 7 + 2𝑥2 − 3𝑥 + 4

= 𝟑𝒙𝟐 + 𝒙 − 𝟑

Make sure your answer is in DESCENDING ORDER!

We can’t SOLVE for the unknown yet, this is as far as we will go in this class.

If you make the Polynomial equal to something, then we can solve: We do this in Grade 10.

Example: (−4𝑝2 + 3 − 2𝑝) + (2 − 3𝑝2 + 7𝑝)

−4𝑝2 + 3 − 2𝑝 + 2 − 3𝑝2 + 7𝑝

−4𝑝2 − 3𝑝2 − 2𝑝 + 7𝑝 + 3 + 2

−𝟕𝒑𝟐 + 𝟓𝒑 + 𝟓

Drop the brackets, leave the sign on

the 1st term of the second Polynomial

Rearrange the terms so like terms are

together, descending order right

away is a bonus

Leave the solution in Descending Order



Example: (5𝑥𝑦 + 3 − 2𝑥) + (−3 + 2𝑥 − 5𝑥𝑦)

5𝑥𝑦 + 3 − 2𝑥 − 3 + 2𝑥 − 5𝑥𝑦

5𝑥𝑦 − 5𝑥𝑦 − 2𝑥 + 2𝑥 + 3 − 3

𝟎

Example: (4 + 3𝑡2 − 7𝑥) + (6𝑥 − 2𝑡2 − 12)

4 + 3𝑡2 − 7𝑥 + 6𝑥 − 2𝑡2 − 12

3𝑡2 − 2𝑡2 − 7𝑥 + 6𝑥 + 4 − 12

𝒕𝟐 − 𝒙 − 𝟖

Subtraction of Polynomials

There is 1 very important concept to understand with subtraction

Consider this: (𝑥2 + 5𝑥 − 4) − (2𝑥2 − 5𝑥 − 4)

We are subtracting this from the 1st one. The subtraction symbol MUST affect each term.

Think about WATERBOMBING in the negative symbol

The signs change

(𝑥2 + 5𝑥 − 4) − (2𝑥2 − 5𝑥 − 4)

After you WATERBOMB in the negative you can change the signs and DROP the BRACKETS

Remember:

o 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 ∗ 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 = 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒

o 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 ∗ 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 = 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒

𝑥2 + 5𝑥 − 4 − 2𝑥2 + 5𝑥 + 4

Drop the brackets, leave the sign on

the 1st term of the second Polynomial

Rearrange the terms so like terms are

together, descending order right

away is a bonus

Leave the solution in Descending Order,

if everything cancels out, zero is a valid

answer!



Now we GROUP the LIKE TERMS and we’re done

𝑥2 − 2𝑥2 = −𝑥2

5𝑥 + 5𝑥 = 10𝑥

−4 + 4 = 0

So, in Descending Order,

−𝒙𝟐 + 𝟏𝟎𝒙

Example: (3𝑥2 − 4𝑥 + 2) − (6𝑥2 + 5𝑥 − 12)

Step 1: Drop the Brackets 3𝑥2 − 4𝑥 + 2 − (6𝑥2 + 5𝑥 − 12)

of 1st Polynomial

Step 2: Waterbomb in the 3𝑥2 − 4𝑥 + 2 − 6𝑥2 − 5𝑥 + 12

(−) to the second one

and Drop the Brackets

Step 3: Group the LIKE TERMS −𝟑𝒙𝟐 − 𝟗𝒙 + 𝟏𝟒

and put the result in

DESCENDING ORDER

Example: (9𝑟2 + 4𝑟 + 5) − (−3𝑟2 − 4𝑟 + 5)

9𝑟2 + 4𝑟 + 5 + 3𝑟2 + 4𝑟 − 5

9𝑟2 + 3𝑟2 + 4𝑟 + 4𝑟 + 5 − 5

𝟏𝟐𝒓𝟐 + 𝟖𝒓

Example: −(2𝑡2 + 4𝑡 − 6) − (8𝑡2 − 5𝑡 + 2)

−2𝑡2 − 4𝑡 + 6 − 8𝑡2 + 5𝑡 − 2

−2𝑡2 − 8𝑡2 − 4𝑡 + 5𝑡 + 6 − 2

−𝟏𝟎𝒕𝟐 + 𝒕 + 𝟒

Waterbomb in the negative sign

Group the LIKE TERMS

Waterbomb in the negative sign Waterbomb in the negative

Group the LIKE TERMS




Add the following Polynomials, leave answer in DESCENDING order.

1. (𝑥 + 4) + (𝑥 − 7)

2. (2𝑥2 − 4𝑥 − 7) + (3𝑥2 − 7 + 4𝑥)

3. (3𝑥𝑦 + 4𝑥3 + 4) + (2𝑥𝑦 − 4𝑥3 − 4) 4. (10 + 4𝑡2 + 4𝑡) + (2𝑡 − 7𝑡2 − 8)

5. (𝑗3 + 2𝑗2 + 𝑗 + 4) + (3𝑗3 − 2𝑗2 − 7𝑗 + 15)

6. (4 + 6𝑥 − 2𝑥2) + (−𝑥2 − 2𝑥) 7. (𝑡2 + 4) + (−𝑡2 − 4)



8. (−𝑥2 + 2 − 3𝑥) + (−4𝑥2 + 𝑥 − 5) 9. (−2𝑥 + 𝑥2 − 2𝑦2) + (−𝑦2 − 𝑥 + 2𝑥2)

10. (4𝑥 − 2𝑥2) + (−5 + 𝑥2) 11. (−3 + 4𝑥2 + 4𝑥) + (5𝑥 − 2𝑥2 + 4)

12. (3𝑥 − 2𝑥𝑦 + 2𝑦) + (𝑥𝑦 − 3𝑦) + (−3𝑦 − 𝑥)

13. (−2𝑦 + 3𝑥 + 𝑥𝑦) + (2𝑥𝑦 − 𝑥 − 𝑦) + (−𝑥 − 4𝑥𝑦)



Subtract the Polynomials, leave answer in DESCENDING order.

14. (3𝑥2 + 4𝑥 − 7) − (−2𝑥2 + 4𝑥 + 9) 15. (𝑡3 − 5𝑡 + 4𝑡2) − (𝑡2 − 7𝑡 − 2𝑡2)

16. (𝑧 − 4) − (3𝑧 − 7) 17. (𝑤 − 7) − (2𝑤 + 4)

18. (𝑟 + 6) − (−2𝑟 − 2) 19. (𝑗 + 14) − (−5𝑗 + 7)

20. (2𝑘2 + 𝑘 − 7𝑘) − (3𝑘2 − 𝑘 − 4𝑘) − (6𝑘2 − 8𝑘 + 7𝑘)



21. (5 − 𝑡) − (−7 + 𝑡) − (12 + 𝑡2) 22. (−2𝑥 − 3𝑦) − (4𝑥 + 2𝑦) − (𝑥 − 3𝑦) 23. (−5𝑥 − 2𝑦 + 3𝑧) − (−2𝑥 + 9𝑦) − (−𝑥 + 𝑦 − 2𝑧)

Perform the Combined Operations

24. (2𝑠𝑡 − 𝑠 − 𝑡) − (−3𝑠𝑡 + 𝑡) + (−𝑠 + 2𝑡) 25. (−3𝑥 + 4𝑦) + (6𝑥 − 5𝑦) − (2𝑥 + 11𝑦 − 5𝑧) 26. (−2𝑥𝑦 + 9𝑧) + (4𝑥2 − 11𝑧) − (6𝑥2 + 8𝑥𝑦 − 11𝑧)



Section 6.3 – Multiply, Divide, Combined Operations, and Tiles

Multiplication of Polynomials

Multiplication is awesome, all we do is WATERBOMB (DISTRIBUTIVITY) and use our

Exponent Laws for the Variables

Remember: When we multiply a COMMON BASE we ADD the exponents!

Also, the Order of Multiplication does not matter!

2 ∗ 3 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒 𝑎𝑠 3 ∗ 2

𝑥𝑦𝑧 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒 𝑎𝑠 𝑦𝑥𝑧, 𝑧𝑦𝑥, 𝑜𝑟 𝑧𝑥𝑦

It makes no difference, keep this in mind

Example: 3(𝑥 + 4)

3(𝑥 + 4) Waterbomb in the 3

3 𝑡𝑖𝑚𝑒𝑠 𝑥 𝑖𝑠 3

3 𝑡𝑖𝑚𝑒𝑠 4 𝑖𝑠 12

So,

𝟑(𝒙 + 𝟒) = 𝟑𝒙 + 𝟏𝟐

Example: −4𝑥(𝑥 + 6)

−4𝑥 ∗ 𝑥 + −4𝑥 ∗ 6

−𝟒𝒙𝟐 − 𝟐𝟒𝒙

Example: 𝑥(𝑥 + 𝑦)

𝑥 ∗ 𝑥 + 𝑥 ∗ 𝑦

𝒙𝟐 + 𝒙𝒚

Example: 4𝑘(3𝑘𝑚 − 2𝑚)

4𝑘 ∗ 3𝑘𝑚 + 4𝑘 ∗ −2𝑚

𝟏𝟐𝒌𝟐𝒎 − 𝟖𝒌𝒎

Example: 𝑡2(6𝑝 − 4𝑡)

𝑡2 ∗ 6𝑝 + 𝑡2 ∗ −4𝑡

6𝑡2𝑝 − 4𝑡3

−𝟒𝒕𝟑 + 𝟔𝒕𝟐𝒑



Example: 14𝑞(2𝑞2 − 4𝑞)

14𝑞 ∗ 2𝑞2 + 14𝑞 ∗ −4𝑞

𝟐𝟖𝒒𝟑 − 𝟓𝟔𝒒𝟐

Division of Polynomials

Division of Polynomials is just fractions and exponent laws

Consider this:

2 + 3

7=

2

7+

3

7

So,

4𝑟 + 2

2=

4𝑟

2+

2

2 → 2𝑟 + 1

Example:

𝑡2 + 7𝑡

𝑡=

𝑡2

𝑡+

7𝑡

𝑡 → 𝑡 + 7

Example:

4𝑧3 − 2𝑧2 + 12𝑧

2𝑧

4𝑧3

2𝑧+

−2𝑧2

2𝑧+

12𝑧

2𝑧

𝟐𝒛𝟐 − 𝒛 + 𝟔

Remember this?

We can break Polynomials down the same way.

𝑡2

𝑡= 𝑡

7𝑡

𝑡= 7

Exponent Laws

Anything divided by itself is 1



Example:

8𝑥𝑦𝑧 + 4𝑦𝑧 + 2𝑧

2𝑧

8𝑥𝑦𝑧

2𝑧+

4𝑦𝑧

2𝑧+

2𝑧

2𝑧

𝟒𝒙𝒚 + 𝟐𝒚 + 𝟏

Combined Operations

It is very rare that you only have to add, subtract, multiply, or divide only.

More often than not it involves a combination of steps

Example:

3𝑟(𝑟 + 4) − 2𝑟(4𝑟 + 6)

3𝑟2 + 12𝑟 − 8𝑟2 − 12𝑟

−5𝑟2

So it’ll take a few steps, multiply first, add/subtract, then combine the terms and leave

your answer in Descending Order

Example:

4𝑡(𝑡2 + 5)

𝑡

4𝑡3 + 20𝑡

𝑡

4𝑡3

𝑡+

20𝑡

𝑡 = 𝟒𝒕𝟐 + 𝟐𝟎

Waterbomb to remove the BRACKETS

Combine LIKE TERMS and SIMPLIFY



Example:

3𝑥(𝑥 + 4)

𝑥+

5𝑥(3𝑥 − 12)

3𝑥

3𝑥2 + 12𝑥

𝑥+

15𝑥2 − 60𝑥

3𝑥

3𝑥2

𝑥+

12𝑥

𝑥+

15𝑥2

3𝑥+

−60𝑥

3𝑥

3𝑥 + 12 + 5𝑥 − 20

𝟖𝒙 − 𝟖

Example:

2𝑟2(𝑟 − 4)

𝑟−

6(𝑟2 + 2𝑟)

2

2𝑟3 − 8𝑟2

𝑟− (

6𝑟2 + 12𝑟

2)

2𝑟3

𝑟−

8𝑟2

𝑟− (

6𝑟2

2+

12𝑟

2)

2𝑟2 − 8𝑟−(3𝑟2 + 6𝑟)

2𝑟2 − 8𝑟 − 3𝑟2 − 6𝑟

2𝑟2 − 3𝑟2 − 8𝑟 − 6𝑟

−𝒓𝟐 − 𝟏𝟒𝒓

Waterbomb to remove brackets

Divide each term by the denominator

Group LIKE TERMS and SIMPLIFY

Waterbomb to remove brackets

Since you’re subtracting, put brackets

around the second Polynomial so you

don’t forget to subtract each term

Divide each term by its denominator

Waterbomb in the NEGATIVE symbol

Group LIKE TERMS

Simplify the final solution



Algebra Tiles: The Visual Representation of Polynomials

If the Tiles are Shaded In, they are the POSTIVE representation, non-shaded are NEGATIVE

That means that the Shaded and Non-Shaded CANCEL OUT

So here are a couple Examples:

Add the following:

𝑥

𝑥 1

𝑥

1 1

1 𝑡𝑖𝑙𝑒 𝑥 𝑡𝑖𝑙𝑒 𝑥2 𝑡𝑖𝑙𝑒

𝑎𝑛𝑑

𝑎𝑛𝑑

𝑎𝑛𝑑 𝑐𝑎𝑛𝑐𝑒𝑙 𝑜𝑢𝑡

𝑐𝑎𝑛𝑐𝑒𝑙 𝑜𝑢𝑡

𝑐𝑎𝑛𝑐𝑒𝑙 𝑜𝑢𝑡

+ =

−𝑥2 − 2𝑥 + 2 𝑥2 + 𝑥 − 3 −𝑥 − 1



Subtract the following (this is tricky):

Multiply the following:

−

Remember when we

illustrated INTEGERS

We need a NEGATIVE to

take away and we don’t

have one so we bring in “0”

Now we TAKE AWAY

=

𝑥2 + 𝑥 − 1 −𝑥2 + 𝑥 − 1 2𝑥2

𝑥(−𝑥 − 2) = −𝑥2 − 2𝑥

−𝑥 − 2

𝑥




Multiply the following. Leave answer in DESCENDING order.

1. −3(𝑥 − 7)

2. −(𝑡2 − 7𝑡 + 4)

3. 4𝑡𝑝(−2𝑡2 + 3𝑝)

4. 4𝑘2(𝑘2 + 7𝑘 − 2)

5. −𝑧(𝑧 + 4)

6. 2𝑥(2𝑦 + 𝑥 − 3𝑧)

7. 𝑥𝑦(𝑥𝑦𝑧 + 𝑧 − 𝑥𝑦) 8. 2𝑠𝑡(−3𝑠 + 4𝑡 − 𝑠𝑡)

9. −2𝑥2(3𝑥2 − 2𝑦2 + 4𝑧2)



Divide the following. Leave answer in DESCENDING order.

10. 3𝑥+12

3 11.

𝑡2+4𝑡

𝑡

12. 3𝑥2−9𝑥+6

3

13. 5𝑞3+10𝑞2−5𝑞

5𝑞 14.

−4𝑡2+2𝑡

2𝑡

15. −𝑎2𝑏𝑐 − 𝑎𝑏2𝑐 + 𝑎𝑏𝑐2

−𝑎𝑏𝑐

16. 18𝑧4−6𝑧3+3𝑧2

−3𝑧2 17. 4𝑟12+6𝑟3−8𝑟2

−2𝑟−2

18. −𝑎2𝑏2𝑐+𝑎𝑏2𝑐2− 𝑎2𝑏2𝑐2

𝑎𝑏2𝑐



Perform the Combined Operations. Answer in DESCENDING order.

19. −3(𝑥2 + 4𝑥) + 5𝑥(𝑥 − 6)

20. 2𝑡(𝑡2−4𝑡)

𝑡 −3𝑡(4𝑡 − 5)

21. 7𝑞(3𝑞2+4𝑞)

7+

9𝑞(6𝑞2−3𝑞)

3

22. −3𝑧3(𝑧−3)

3−

4𝑧2(3𝑧+6𝑧2)

3



23. Add

24. Multiply

+ −𝑥2 𝑥2 −𝑥 −𝑥

−1 +1



Extra Work Space



Answer Key

Section 6.1

1. 𝑇𝑒𝑟𝑚𝑠: 3 3𝑥, −4𝑥2, −5

𝐷𝑒𝑔𝑟𝑒𝑒: 1, 2, 0

2. 𝑇𝑒𝑟𝑚𝑠: 1 4𝑥𝑦𝑧

𝐷𝑒𝑔𝑟𝑒𝑒: 3

3. 𝑇𝑒𝑟𝑚𝑠: 3 −2𝑥𝑦𝑧, −5𝑥𝑦, 4 𝐷𝑒𝑔𝑟𝑒𝑒: 3, 2, 0

4. 𝑇𝑒𝑟𝑚𝑠: 3 5𝑥3𝑦, 4𝑥𝑦3, −6𝑥𝑦𝑧

𝐷𝑒𝑔𝑟𝑒𝑒: 4, 4, 3

5. 𝑇𝑒𝑟𝑚𝑠: 1 5

𝐷𝑒𝑔𝑟𝑒𝑒: 0

6. 𝑇𝑒𝑟𝑚𝑠: 4 3𝑥, 4𝑦, 5𝑧, −𝑥2

𝐷𝑒𝑔𝑟𝑒𝑒: 1, 1, 1, 2

7. 4𝑥2 − 5𝑥 + 3 8. 4𝑡3 − 3𝑡2 − 4𝑡

9. 5𝑥2 − 𝑥 + 2 10. 𝑧2 − 4𝑧 + 5 11. 𝑥𝑦 + 𝑥𝑧 + 𝑥 − 𝑦 12. 𝑥2 − 5𝑥𝑦 + 2𝑥 − 𝑦

13. −𝑡2 − 3𝑡 + 8 14. 7𝑧3 − 5𝑧2 + 5𝑧 − 4 15. 5 16. 0

17. 1

6𝑖2 + 2𝑖 − 5 18. −6.2𝑥 + 𝑦 + 1 19.

8

5𝑥 +

1

3𝑦 + 10 20.

15

16𝑗2 −

3

2𝑗

Section 6.2

1. 2𝑥 − 3 2. 5𝑥2 − 14 3. 5𝑥𝑦 4. −3𝑡2 + 6𝑡 + 2

5. 4𝑗3 − 6𝑗 + 19 6. −3𝑥2 + 4𝑥 + 4 7. 0 8. −5𝑥2 − 2𝑥 − 3

9. 3𝑥2 − 3𝑦2 − 3𝑥 10. −𝑥2 + 4𝑥 − 5 11. 2𝑥2 + 9𝑥 + 1 12. −𝑥𝑦 + 2𝑥 − 4𝑦

13. −𝑥𝑦 + 𝑥 − 3𝑦 14. 5𝑥2 − 16 15. 𝑡3 + 5𝑡2 + 2𝑡 16. −2𝑧 + 3

17. −𝑤 − 11 18. 3𝑟 + 8 19. 6𝑗 + 7 20. −7𝑘2

21. −𝑡2 − 2𝑡 22. −7𝑥 − 2𝑦 23. −2𝑥 − 12𝑦 + 5𝑧 24. 5𝑠𝑡 − 2𝑠

25. 𝑥 − 12𝑦 + 5𝑧 26. −2𝑥2 − 10𝑥𝑦 + 9𝑧

Section 6.3

1. −3𝑥 + 21 2. −𝑡2 + 7𝑡 − 4 3. −8𝑡3𝑝 + 12𝑡𝑝2 4. 4𝑘4 + 28𝑘3 − 8𝑘2

5. −𝑧2 − 4𝑧 6. 2𝑥2 + 4𝑥𝑦 − 6𝑥𝑧 7. 𝑥2𝑦2𝑧 − 𝑥2𝑦2 + 𝑥𝑦𝑧 8. −2𝑠2𝑡2 − 6𝑠2𝑡 + 8𝑠𝑡2

9. −6𝑥4 + 4𝑥2𝑦2 − 8𝑥2𝑧2 10. 𝑥 + 4 11. 𝑡 + 4 12. 𝑥2 − 3𝑥 + 2

13. 𝑞2 + 2𝑞 − 1 14. −2𝑡 + 1 15. 𝑎 + 𝑏 − 𝑐 16. −6𝑧2 + 2𝑧 − 1

17. −2𝑟14 − 3𝑟5 + 4𝑟4 18. −𝑎𝑐 − 𝑎 + 𝑐 19. 2𝑥2 − 42𝑥 20. −10𝑡2 + 7𝑡

21. 21𝑞3 − 5𝑞2 22. −9𝑧4 − 𝑧3 23. −4𝑥 − 1 24. 2𝑥

Documents

Section 6: Polynomials