MathematicsQuarter 1 – Module 8:Performing Division of

Polynomials Using Long andSynthetic Division

10

Mathematics – Grade 10Alternative Delivery ModeQuarter 1 – Module 8 : Performs division of polynomials using long and

synthetic divisionFirst Edition, 2020

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Published by the Department of EducationSecretary: Leonor Magtolis BrionesUndersecretary: Diosdado M. San Antonio

Printed in the Philippines by:

Department of Education – Cordillera Administrative Region

Office Address: Wangal, La Trinidad, BenguetTelefax: (074) 422-4074E-mail Address: car@deped.gov.ph

Development Team of the Module

Writer’s Name: Laila B. Kiw-isenEditor’s Name: Melchor B. TicagReviewer’s Name: Bryan A. HidalgoManagement Team:

May B. Eclar, PhDMarie Carolyn B. VeranoCarmel F. MerisEthielyn E. TaquedEdgar H. MadlaingSoraya T. FaculoFrancisco C. Copsiyan

10MathematicsQuarter 1 – Module 8:

Performs division of polynomialsusing long and synthetic

divisionM10AL-lg-1

0

ii

Introductory MessageThis is the eight learning competency in our Mathematics 10 curriculumstandards hence mastery of the skills is significant to have a smoothprogress in the succeeding lessons.

This module was collaboratively designed, developed and reviewed byeducators both from public and private institutions to assist you, theteacher or facilitator in helping the learners meet the standards set by the Kto 12 Curriculum while overcoming their personal, social, and economicconstraints in schooling.

For the facilitator:This module is intended to help grade 10 students understand and masterthe concepts of dividing polynomials. It is designed to equip them withessential knowledge about the said topic and skills on solving problemsinvolving polynomials. Please have patience in assisting the learnersaccomplish this module.

For the learner:

This learning material provides you with set of activities that will help youunderstand how to divide polynomials. By doing the prepared activities, it isexpected that you will develop your skill in dividing polynomials that is avery important skill on the succeeding lessons. This module was alsodesigned so that you can pace your own learning as you achieve therequired learning competencies. Please read completely the written texts andfollow the instructions carefully so that you will be able to get the most ofthis learning material. We hope that you will enjoy learning!

Here is your guide on the parts of the learning modules which you need tounderstand as you progress in reading and analyzing its content.

ICON LABEL DETAIL

What I need to knowThis will give you an idea of the skillsor competencies you are expected tolearn in the module.

What I know

This part includes an activity thataims to check what you already knowabout the lesson to take. If you get allthe answers correct (100%), you maydecide to skip this module.

iii

What’s inThis is a brief drill or review to helpyou link the current lesson with theprevious one.

What’s new

In this portion, the new lesson will beintroduced to you in various wayssuch as a story, a song, a poem, aproblem opener, an activity or asituation.

What is itThis section provides a briefdiscussion of the lesson. This aims tohelp you discover and understandnew concepts and skills.

What’s more

This comprises activities forindependent practice to solidify yourunderstanding and skills of the topic.You may check the answers to theexercises using the Answer Key at theend of the module.

What I have learnedThis includes questions or blanksentence/paragraph to be filled in toprocess what you learned from thelesson.

What I Can do

This section provides an activitywhich will help you transfer your newknowledge or skill into real lifesituations or concerns.

AssessmentThis is a task which aims to evaluateyour level of mastery in achieving thelearning competency.

Additional Activities

In this portion, another activity will begiven to you to enrich your knowledgeor skill of the lesson learned. Thisalso tends retention of learnedconcepts.

Answer KeyThis contains answers to all activitiesin the module.

At the end of this module you will also find:

References This is a list of all sources used indeveloping this module.

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The following are some reminders in using this module:

1. Use the module with care. Do not put unnecessary mark/s on anypart of the module. Use a separate sheet of paper in answering theexercises.

2. Don’t forget to answer What I Know before moving on to the otheractivities included in the module.

3. Read the instruction carefully before doing each task.4. Observe honesty and integrity in doing the tasks and checking your

answers.5. Finish the task at hand before proceeding to the next.6. Return this module to your teacher/facilitator once you are through

with it.

If you encounter any difficulty in answering the tasks in this module, do nothesitate to consult your teacher or facilitator. Always bear in mind that youare not alone.

We hope that though this material, you will experience meaningful learningand gain deep understanding of the relevant competencies. You can do it!

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What I Need to Know

This module was designed and written with you in mind. It is here toindulge you in dividing polynomials using long method and syntheticdivision. The scope of this module permits it to be used in many differentlearning situations. The arrangement of the lessons follows the standardsequence of the course. But the pacing in which you read and answer thismodule is dependent on your ability.

After going through this module, you are expected to:

1) divide polynomials using long division and synthetic division,and

2) solve word problem that involves dividing polynomials.

What I Know

DIRECTION: Choose the letter of your answer from the given choices. Writeyour answer on a separate sheet of paper.

1) If a fifth degree polynomial is divided by a third degree polynomial, what

is the degree of the quotient?

A) 1 B) 2 C) 3 D) 4

2) Divide: �� � ��� � �� � 慮 by �� � �.

A) �� � 湉�� 慮 B)�� � �� � � C) � � � D) � � �

2

3) In the division algorithm)()()(

)()(

xdxrxQ

xdxP

, what is the dividend?

A) �)�⺂ B) �)�⺂ C) �)�⺂ D) �)�⺂

4) Find the remainder when �� � ��� � t�� � is divided by � � ��

A) �湉 B) � C) � � D) � �湉

5) If �� � ��� � ��� 慮 is divided by � � , the remainder is zero.

A) True B) False C) Cannot be determined

For #6 to 8, use the illustration of long division below:

Divide )��� � ��� � � � �⺂ �t )� � t⺂

162

)16441( 214

41 103

4010 10 123

223 4

2

2

2

23

23

xx

xx

xx-xx

xxxxxx

6) What is the remainder?

A) t� � 慮t B) � � t C) t D) 162

7) Which is the divisor?

A) � � t B) �� � 慮 C) ��� � 慮� � t D) 162

8) Which is the quotient?

A) � � t B) �� � 慮 C) ��� � 慮� � t D) 162

3

9) Using synthetic division to divide �t� � 湉t� � �慮 by t � � , which of thefollowing is the correct first row of the algorithm?

A) � � � � 湉 � �慮 C) � � � � 湉 慮 � �慮B) � � � 湉 � �慮 D) � � � 湉 慮 � �慮

For #s 10 to 11, use the following synthetic division:

10) Which polynomial is the dividend?

A) �t � 慮�� � ��� � �� B) �� � 慮�� � �� � �C) �� � t�� � 慮� D) �� � t�� 慮

11) Which polynomial is the quotient?

A) �t � 慮�� � ��� � �� B) �� � 慮�� � �� � �C) �� � t�� � 慮� D) �� � t�� 慮

12) Find the quotient when �� � ��� � � � �慮 is divided by � � � .

A) �� � �� � t B) �� � �� � tC) �� � �� � t D) �� � �� � t

13) What is the quotient when ��� � ㌳�� � �� � � is divided by )�� � )?

A) �� � t�� � B) ��� � �� � 慮C) �� � t�� � D) ��� � �� � 慮

14) What will be multiplied to �� � 慮�� � to get ��� � ㌳�� � �� � �?

A) �� � B) �� � C) � � � D) �� � �

15) The volume of a rectangular prism is )��� � �� � �� � t⺂��� and itsheight is )� � t⺂ ��. What is the area of its base?

A) )��� � �� � ⺂ ��� B) )��� � ㌳� � 湉慮⺂ ���

C) )��� � �� � ⺂ ��� D) )��� � ㌳� � 湉慮⺂ ���

�2 1 6 2 �12�2 � � �

1 t � 慮 0

4

LESSON

1Division of PolynomialsUsing Long and SyntheticDivision

What’s In

When you were in Grade 7, you had learned that polynomial is an algebraicexpression whose variables must have nonnegative-integer powers. Thedegree of a polynomial in one variable is the highest exponent among all theterms in the polynomial. Recall also the concepts about naming polynomialsaccording to its number of terms and its degree. Then answer the followingactivity (the first polynomial was done for you) in a separate answer sheet.

Polynomial Numberof Terms

Name of thePolynomial

(according to thenumber of terms)

DegreeName of thePolynomial

(according to degree)

Example: �� � �� � 3 Trinomial 2 Quadratic1) � � 2) ��t � ��� � t�� 3) � ���

4) �慮 � ��� � 5) 3�� � ��

5

What’s New

Some real life applications of polynomials can be seen in the fieldengineering and economy. Engineers used polynomials when designingroads, buildings and other structures and economist used it to modeleconomic growth pattern. To understand more about polynomials, let’s havethe following problem:

Problem:

During a school fund raising activity, you were able to generate a totalof �� � ㌳� � ��� � �湉 pesos from selling candies. If each candy costs� � � pesos, how many candies were you able to sell?

To find the number of candies, we can use dividing polynomials which is thefocus of this module.

What Is It

In this part of the module, we will show you how to divide polynomials usinglong division and synthetic division. This skill is very important in factoringand solving for the roots of a polynomial equation.

A. Dividing Polynomials using Long Division

To show you how to divide polynomials using long division, we will use thepolynomials specified in the problem from the previous page. That is todivide �� � ㌳� � ��� � �湉 by � � �. Follow the steps:

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Example 1. Divide )�� � ㌳� � ��� � �湉⺂ by )� � �⺂.

Solution:

1) Arrange the polynomials in descending powers of � and write in the formdividenddivisor . If there are missing terms, replace it with 0.

32739 23

xxxx 27933 23 xxxx

2) Divide the first term of the dividend by the first term of the divisor to getthe first term of the quotient:

27933

2

23x

xxxx

3) Multiply the divisor by the first term of the quotient.

327933

2

23

23x

xxxxxx

4) Subtract the product from the dividend then bring down the next term.

96 3

27933

2

2

23

23x

xxxx

xxxx

5) Repeat steps 2 to 4. This time, the difference and the next term will bethe new dividend.

6

279

186 96

327933

2

2

2

23

23xx

x

xxxx

xxxxxx

�� � � ᔖ ��

� � � �� ᔖ �� � ���

�

�

� 慮�� � � ᔖ� 慮�

� � � � 慮� ᔖ� 慮�� � ��

7

6) Continue the process until a remainder is obtained. The remainder canbe zero or a polynomial whose degree is lower than the divisor.

96

0

279 279

186 96

327933

2

2

2

23

23

xx

xx

xxxx

xxxxxx

Therefore, the quotient is �� � 慮�� ㌳.

In general, when a polynomial is divided by another polynomial, we expressthe result in the following form:

�)�⺂�)�⺂

ᔖ � � ��)�⺂�)�⺂

Where �)�⺂ is the dividend, �)�⺂ � 慮 is the divisor, �)�⺂ is the quotient,and �)�⺂ is the remainder.

�

㌳� � � ᔖ ㌳

� � � ㌳� ᔖ ㌳� � �湉

8

Example 2. Divide )�t � ��� � t�� � 慮� � �⺂ by )�� � �⺂.

Solution:

Follow the procedures shown from example 1.

10

5 5

)10 2( 10 2

)5 (

12510425

2

2

3

23

24

2

2342

xx

xxxxx

xx

xxxxxxx

51012

551042

22

2

234

x

xxx

xxxx

Quotient

Remainder

Multiply: ��)�� � �⺂

Subtract. Bring down � 慮�

Multiply: ��)�� � �⺂

Subtract. Bring down �

Multiply: )�� � �⺂

9

LESSON

2Dividing PolynomialsUsing Synthetic Division

There is a more efficient way of dividing polynomials if the divisor is a linearbinomial in the form � � h . This method is called synthetic division. Adetailed discussion on how this synthetic division will be done is given below.Just like long division, arrange the polynomials first in descending powers of� and write 0 as coefficient of any missing term.

Example 3: Divide )�� � ㌳� � ��� � �湉⺂ by )� � �⺂ using synthetic division.

(These polynomials are the same as example 1. Let’s find out if synthetic andlong division will give the same quotient).

Arrange the dividend in descending order: �� � ��� � ㌳� � �湉

�3 1 �3 �9 27 Write the coefficient of thepolynomials.The divisor is � � �, so use h ᔖ� �

�3 1 �3 �9 27

1 Bring down the first coefficient

�3 1 �3 �9 27�3 Multiply � � by 1 and write the result

below � �1

�3 1 �3 �9 27�3

1 � 慮 Add � � and � �

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�3 1 �3 �9 27�3 18 �27 Multiply � � by � 慮 and write the result below � ㌳

1 �6 9 0 Add � ㌳ and �. Repeat the process until allcolumns are filled

�3 1 �3 �9 27 The degree of the quotient is one less than�3 18 �27 the degree of the dividend.

1 �6 9 0 Thus, the quotient is �� � 慮�� ㌳�

In this example, we were able to see that both long division and syntheticdivision yield the same answer.

Example 4: Divide )��t � ��� � � � t⺂ by )� � �⺂

� 2 0 � � 1 �4

Write the coefficient of the polynomials. Since �� is a

missing term, write 0 as its coefficient. The divisor is� � �, so use h ᔖ �

� 2 0 � � 1 �4

Perform the synthetic division.

4 8 10 222 4 5 11 18

� 2 0 � � 1 �4

Identify the quotient and the remainder.

4 8 10 222 4 5 11 18

21811542

2432 23

24

x

xxxx

xxx

Coefficient ofthe quotient

remainder

Coefficient ofthe quotient

remainder

11

Example 5: Divide )��� � 慮�� � �� � �⺂ by )�� � �⺂

In this example, the leading coefficient of the divisor is not 1. Hence, divideboth terms of the divisor by 3 so that it will be in the form � � h. So the newdivisor will now be � � �

�.

���

3 �16 � 12 Write the coefficient of the polynomials.

The divisor is � � ��, so use h ᔖ� �

�.

���

3 �16 � 12 Perform the synthetic division.

� � 12 �103 � � 15 �

���

3 �16 � 12 Identify the quotient and the remainder.

� � 12 �103 � � 15 �

Quotient: ��� � �� � �Remainder: 2

Since the divisor �� � � was divided by 3 to get h ᔖ� ��, then divide also

every terms in the quotient (except remainder) by 3. Hence, the quotient willbe:

�� � 慮�� �.

23256

23123163 2

23

x

xxx

xxx

Coefficient ofthe quotient

remainder

12

What’s More

Now, your turn!

Activity 1: Divide the following polynomials using long division.

a) )��� � ㌳�� � �� � t⺂ � )� � t⺂

b) )��� � �湉⺂ � )�� � �⺂

c)2

14122011 32

xxxx

Activity 2: Divide the following polynomials using synthetic division

a) t�� � � � � � )�� �⺂

b) ��� � t�� �t � ��� � t � � � �

c) ��� � ��� � t�� � � � � �

d) � �t � ��� � �� � ��� � � � � �

e) ��� � ��� � t�� � � �� �

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What I Have Learned

Let us sum up what we had learned in this module.

A) Fill in the blank with the correct term. Write your answer on aseparate sheet of paper.

When dividing polynomials, we express the result in the following form:�)�⺂�)�⺂ ᔖ � � �

�)�⺂�)�⺂

Where �)�⺂ is the 1 , �)�⺂ � 慮 is the 2 ,�)�⺂ is the 3 , and �)�⺂ is the 4 .

B) Use the synthetic division below to find the following:

2 2 0 0 �1

�36

a� Dividend:

4 8 16 30 b� Divisor:2 4 8 15 �6 c. Remainder:

d. Quotient:

14

What I Can Do

In this activity, you will apply dividing polynomial in solving word problems.

Solve the following problems:

1) Pedro bought ��� � �㌳�� � �� � 慮 pesos worth of tokens foryour classroom Christmas party. If each token is worth �� � tpesos, how many tokens did Peter buy?

2) Lita works for )� � �⺂ hours as a service crew in a fast-foodchain. She earns ��� � ���� � �慮� � � pesos today. How muchdoes Lita earn per hour?

3) The area of a rectangle is t�t � t�� � t�� � �� � squarecentimeter. If the length of the rectangle is ��� � centimeter,what is the width of the rectangle?

15

Assessment

DIRECTION: Choose the letter of your answer from the given choices. Writeyour answer on a separate sheet of paper.

1) If a sixth degree polynomial is divided by a second degree polynomial,what is the degree of the quotient?

A) 1 B) 2 C) 3 D) 4

2) Divide: �� � ��� � �� � 慮 by �� � �.

A) �� � �� � 慮 B) �� � 湉�� �慮 C) � � � D) � � �

3) In the division algorithm)()()(

)()(

xdxrxQ

xdxP

, what is the divisor?

A) �)�⺂ B) �)�⺂ C) �)�⺂ D) �)�⺂

4) Find the remainder when �� � ��� � t�� � is divided by � � ��

A) �湉 B) � C) � � D) � �湉

5) If �� � ��� � ��� 慮 is divided by � � , the remainder is zero.

A) True B) False C) Cannot be determined

16

For #s 6 to 8, use the illustration of long division below:

Divide )��� � ��� � 慮� � �⺂ �t )� � �⺂

63

)5117( 1217

17 92

279109

621210323

2

2

2

23

23

xx

xx

xx-xx

xxxxxx

6) What is the remainder?

A) 17 B) 63 C) � � � D) 湉� � �

7) Which is the divisor?

A) ��� � ㌳� � 湉 B) � � � C) ��� � ��� � 慮� � � D) 63

8) Which is the quotient?

A) ��� � ㌳� � 湉 B) � � � C) ��� � ��� � 慮� � � D) 63

9) Using synthetic division to divide 湉t� � �t� � �慮 by t � � , which of thefollowing is the correct first row of the algorithm?

A) � 湉 � � 慮 � �慮 C) � � 湉 � � � �慮B) � � 湉 � � 慮 � �慮 D) � 湉 � � � �慮

17

For #s 10 to 11, use the following synthetic division:

3 1 �6 2 213 � ㌳ � �

1 � � � 湉 0

10) Which polynomial is the dividend?

A) �� � 慮�� � �� � � B) �t � 慮�� � ��� � ��C) �� � ��� � 湉� D) �� � �� � 湉

11) Which polynomial is the quotient?

A) �� � 慮�� � �� � � B) �t � 慮�� � ��� � ��C) �� � ��� � 湉� D) �� � �� � 湉

12) Find the quotient when �� � ��� � ㌳� � �慮 is divided by � � � .

A) �� � �� � t B) �� � �� � tC) �� � �� � t D) �� � �� � t

13) What is the quotient when ��� � ㌳�� � �� � � is divided by )�� � )?

A) �� � t�� � B) �� � t�� C) ��� � �� � 慮 D) ��� � �� � 慮

14) What will be multiplied to �� � 慮�� � to get ��� � 湉�� � �?

A) � � � B) �� � � C) �� � D) �� �

15) The volume of a rectangular prism is )��� � �� � � � t⺂��� and itsheight is )� � t⺂ ��. What is the area of its base?

A) )��� � �� � ⺂ ��� B) )��� � ㌳� � 湉慮⺂ ���

C) )��� � �� � ⺂ ��� D) )��� � ㌳� � 湉慮⺂ ���

18

Additional Activity

Answer the following problems:

A) Complete the synthetic division given the polynomial

��t � ��� � �� � t divided by � � ��

2 �3 �2 44 4 4

2 2 2

B) The long division below is incorrect. What mistake was made?

Divide: �� � 慮�� � �� � � � � � �

17

)2010( 310

10 4

8424-

23262

2

2

2

23

23

xx

xx

xx-xx

xxxxxx

C) Find the constant � such that the denominator will divide evenlyinto the numerator.

1)4

92 23

xCxxx

2)3

33 234

x

+Cx −xxx

19

Answer Key

20

REFERENCES:

Algebraic Division taken from http://mathematics.laerd.com

Larson, R. (2006). Precalculus with limits. Boston, MA: Houghton Mifflin Co.

Leithold, L. (1989). College Algebra and Trigonometry. Reading Mass.:

Addison – Wesley.

Young, C. (2010). Precalculus. Hoboken, N.J.: Wiley.

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