MTH-4106-1 F actoring - SOFAD · PDF file3. factoring trinomials of the form ax2 + bx + c or ax2 + bxy + cy2; 5. factoring differences of squares. To factor a polynomial is to write

F actoring

and

Algebraic Fractions

MTH-4106-1

MTH-4106-1 C1-C4 Factorization 1/31/12 11:38 AM Page 1

FACTORING

AND

ALGEBRAIC

FUNCTIONS

MTH-4106-1

Project Coordinator: Jean-Paul Groleau

Authors: Nicole PerreaultSuzie Asselin

Content Revision: Jean-Paul GroleauAlain Malouin

Updated Version: Line Régis

Translator: Claudia de Fulviis

Linguistic Revision: Johanne St-Martin

Desktop Publishing: P.P.I. inc.

Cover Page: Daniel Rémy

First Printing: 2005

Printing: 2005

Reprint: 2006

© Société de formation à distance des commissions scolaires du Québec

All rights for translation, adaptation, in whole or in part, reserved for all countries. Anyreproduction by mechanical or electronic means, including microreproduction, isforbidden without the written permission of a duly authorized representative of theSociété de formation à distance des commissions scolaires du Québec (SOFAD).

Legal Deposit — 2005

Bibliothèque et Archives nationales du Québec

Bibliothèque et Archives Canada

ISBN 978-2-89493-284-1

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MTH-4106-1 Factoring and Algebraic Fractions

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TABLE OF CONTENTS

Introduction to the Program Flowchart ................................................... 0.4The Program Flowchart ............................................................................ 0.5How to Use This Guide ............................................................................. 0.6General Introduction................................................................................. 0.9Intermediate and Terminal Objectives of the Module ............................ 0.11Diagnostic Test on the Prerequisites ....................................................... 0.15Answer Key for the Diagnostic Test on the Prerequisites ...................... 0.19Analysis of the Diagnostic Test Results ................................................... 0.21Information for Distance Education Students......................................... 0.23

UNITS

1. Factoring by Removing the Common Factor ........................................... 1.12. Factoring by Grouping .............................................................................. 2.13. Factoring Trinomials of the Form x2 + bx + c

or x2 + bxy + cy2 ......................................................................................... 3.14. Factoring Trinomials of the Form ax2 + bx + c

or ax2 + bxy + cy2 ....................................................................................... 4.15. Factoring Differences of Two Squares ..................................................... 5.16. Factoring Polynomials .............................................................................. 6.17. Simplifying Algebraic Fractions ............................................................... 7.18. Multiplying and Dividing Two Algebraic Fractions ................................ 8.19. Adding and Subtracting Two Algebraic Fractions and Comparing

Algebraic Expressions ............................................................................... 9.1

Final Review ............................................................................................ 10.1Answer Key for the Final Review........................................................... 10.6Terminal Objectives ................................................................................ 10.8Self-Evaluation Test................................................................................ 10.11Answer Key for the Self-Evaluation Test .............................................. 10.17Analysis of the Self-Evaluation Test Results ........................................ 10.21Final Evaluation...................................................................................... 10.22Answer Key for the Exercises ................................................................. 10.23Glossary ................................................................................................... 10.77List of Symbols ........................................................................................ 10.82Bibliography ............................................................................................ 10.83

Review Activities ..................................................................................... 11.1

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INTRODUCTION TO THE PROGRAM FLOWCHART

Welcome to the World of Mathematics!

This mathematics program has been developed for the adult students of the

Adult Education Services of school boards and distance education. The learning

activities have been designed for individualized learning. If you encounter

difficulties, do not hesitate to consult your teacher or to telephone the resource

person assigned to you. The following flowchart shows where this module fits

into the overall program. It allows you to see how far you have progressed and

how much you still have to do to achieve your vocational goal. There are several

possible paths you can take, depending on your chosen goal.

The first path consists of modules MTH-3003-2 (MTH-314) and MTH-4104-2

(MTH-416), and leads to a Diploma of Vocational Studies (DVS).

The second path consists of modules MTH-4109-1 (MTH-426), MTH-4111-2

(MTH-436) and MTH-5104-1 (MTH-514), and leads to a Secondary School

Diploma (SSD), which allows you to enroll in certain Cegep-level programs that

do not call for a knowledge of advanced mathematics.

The third path consists of modules MTH-5109-1 (MTH-526) and MTH-5111-2

(MTH-536), and leads to Cegep programs that call for a solid knowledge of

mathematics in addition to other abiliies.

If this is your first contact with this mathematics program, consult the flowchart

on the next page and then read the section “How to Use This Guide.” Otherwise,

go directly to the section entitled “General Introduction.” Enjoy your work!

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CEGEP

MTH-5110-1 Introduction to Vectors

MTH-5109-1 Geometry IV

MTH-5108-1 Trigonometric Functions and Equations

MTH-5107-1 Exponential and Logarithmic Functions and Equations

MTH-5106-1 Real Functions and Equations

MTH-5105-1 Conics

MTH-5104-1 Optimization II

MTH-5103-1 Probability II

MTH-5102-1 Statistics III

MTH-5101-1 Optimization I

MTH-4110-1 The Four Operations on Algebraic Fractions

MTH-4109-1 Sets, Relations and Functions

MTH-4108-1 Quadratic Functions

MTH-4107-1 Straight Lines II

MTH-4106-1 Factoring and Algebraic Functions

MTH-4105-1 Exponents and Radicals

MTH-4103-1 Trigonometry I

MTH-4102-1 Geometry III

MTH-536

MTH-526

MTH-514

MTH-436

MTH-426

MTH-416

MTH-314

MTH-216

MTH-116

MTH-3002-2 Geometry II

MTH-3001-2 The Four Operations on Polynomials

MTH-2008-2 Statistics and Probabilities I

MTH-2007-2 Geometry I

MTH-2006-2 Equations and Inequalities I

MTH-1007-2 Decimals and Percent

MTH-1006-2 The Four Operations on Fractions

MTH-1005-2 The Four Operations on Integers

MTH-5111-2 Complement and Synthesis II

MTH-4111-2 Complement and Synthesis I

MTH-4101-2 Equations and Inequalities II

MTH-3003-2 Straight Lines I

TradesDVS

MTH-5112-1 Logic

25 hours = 1 credit

50 hours = 2 credits

MTH-4104-2 Statistics II

PROGRAM FLOWCHART

You ar e here

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Hi! My name is Monica and I have beenasked to tell you about this math module.What’s your name?

I’m Andy.

Whether you areregistered at anadult educationcenter or pur-suing distanceeducation, ...

You’ll see that with this method, math isa real breeze!

... you have probably taken aplacement test which tells youexactly which module youshould start with.

My results on the testindicate that I should beginwith this module.

Now, the module you have in yourhand is divided into threesections. The first section is...

... the entry activity, whichcontains the test on theprerequisites.

By carefully correcting this test using thecorresponding answer key, and record-ing your results on the analysis sheet ...

HOW TO USE THIS GUIDE

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?

The memo pad signals a brief reminder ofconcepts which you have already studied.

The calculator symbol reminds you thatyou will need to use your calculator.

The sheaf of wheat indicates a review designed toreinforce what you have just learned. A row ofsheaves near the end of the module indicates thefinal review, which helps you to interrelate all thelearning activities in the module.

The starting lineshows where thelearning activitiesbegin.

The little white question mark indicates the questionsfor which answers are given in the text.?

... you can tell if you’re well enoughprepared to do all the activities in themodule.

The boldface question markindicates practice exerciseswhich allow you to try out whatyou have just learned.

And if I’m not, if I need a littlereview before moving on, whathappens then?

In that case, before you start theactivities in the module, the resultsanalysis chart refers you to a reviewactivity near the end of the module.

In this way, I can be sure Ihave all the prerequisitesfor starting.

Exactly! The second sectioncontains the learning activities. It’sthe main part of the module.

Look closely at the box tothe right. It explains thesymbols used to identify thevarious activities.

The target precedes theobjective to be met.

Good!

?

START

Lastly, the finish line indicatesthat it is time to go on to the self-evaluationtest to verify how well you have understoodthe learning activities.

FINISH

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A “Did you know that...”?

Later ...

For example. words in bold-face italics appear in theglossary at the end of themodule...

Great!

... statements in boxes are importantpoints to remember, like definitions, for-mulas and rules. I’m telling you, the for-mat makes everything much easier.

The third section contains the final re-view, which interrelates the differentparts of the module.

Yes, for example, short tidbitson the history of mathematicsand fun puzzles. They are in-teresting and relieve tension atthe same time.

No, it’s not part of the learn-ing activity. It’s just there togive you a breather.

There are also many fun thingsin this module. For example,when you see the drawing of asage, it introduces a “Did youknow that...”

Must I memorize what the sage says?

It’s the same for the “math whiz”pages, which are designed espe-cially for those who love math.

They are so stimulating thateven if you don’t have to dothem, you’ll still want to.

And the whole module hasbeen arranged to makelearning easier.

There is also a self-evaluationtest and answer key. They tellyou if you’re ready for the finalevaluation.

Thanks, Monica, you’ve been abig help.

I’m glad! Now,I’ve got to run.

See you!This is great! I never thought that I wouldlike mathematics as much as this!

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GENERAL INTRODUCTION

In this module, we will look at factoring and the four operations (+, –, ×, ÷)

on algebraic fractions.

In the first part of the module, you will learn the five factoring methods:

1. factoring by removing the common factor;

2. factoring by grouping;

3. factoring trinomials of the form x2 + bx + c or x2 + bxy + cy2;

3. factoring trinomials of the form ax2 + bx + c or ax2 + bxy + cy2;

5. factoring differences of squares.

To factor a polynomial is to write the polynomial as a product of two or more

polynomials. In other words, to factor a polynomial is to find the factors of

the polynomial. Each of these methods will be examined in a separate unit.

Factoring is a precious mathematical tool for solving second-degree equations,

i.e., equations in which the highest exponent is 2.

Unfortunately, certain polynomials are not factorable. To solve equations

containing this type of polynomial, it is necessary to resort to more advanced

techniques which will be covered in a subsequent module.

In the second part of the module, you will learn how to perform various

operations on algebraic fractions. You will first learn how to simplify them

(factoring the numerator and the denominator). It is important to master this

skill before going on, for you will use it in all the units in the second part of the

module. Indeed, all your results will have to be reduced to lowest terms.

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In the units that follow the unit on simplification, you will learn how to multiply,

divide, add and subtract algebraic fractions.

What you have already learned about the operations on numerical fractions will

help you a great deal here. Furthermore, a keen sense of observation, order and

method will definitely come in handy.

These are the main concepts that will be covered in this module on factoring and

the four operations on algebraic fractions.

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INTERMEDIATE AND TERMINAL OBJECTIVES OFTHE MODULE

Module MTH-4106-1 consists of nine units and requires 25 hours of study

distributed as shown below. The terminal objectives appear in boldface.

Objectives Number of hours** % (evaluation)

1 to 6 11 35%

7 to 9 13 65%

* One hour is allotted for the final evaluation.

1. Factoring by removing the common factor

Find the common factor of all the terms of a polynomial containing up to six

terms linked by + or – signs. The result must be expressed as the product of

a monomial and a polynomial, which is placed in parentheses. The numerical

coefficients of the terms of the polynomial are rational numbers, and the

exponents of the variables are natural numbers.

2. Factoring by grouping

Factor a polynomial of up to six terms linked by + or – signs by applying the

method of grouping. The result must be expressed as the product of two

binomials or as the product of a binomial and a trinomial. The terms of the

polynomial may have to be rearranged before being grouped and factored.

The numerical coefficients of the terms of the polynomial are rational

numbers, and the exponents of the variables are natural numbers. The steps

in the solution must be shown.

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3. Factoring trinomials of the form x2 + bx + c or x2 + bxy + cy2

Factor a trinomial of the form x2 + bx + c or x2 + bxy + cy2, where b and c are

integers. The result must be expressed as the product of two binomials of the

form (x + d)(x + e) or (x + dy)(x + ey), where d and e are integers. The steps in

the solution must be shown.

4. Factoring trinomials of the form ax2 + bx + c or ax2 + bxy + cy2

Factor a trinomial of the form ax2 + bx + c or ax2 + bxy + cy2, where a, b and

c are integers. The result must be expressed as the product of two binomials

of the form (kx + l)(mx + n) or (kx + ly)(mx + ny), where k, l, m and n are

integers. The steps in the solution must be shown.

5. Factoring differences of two squares

Factor the difference of two squares as the product of two binomials

consisting of the sum and the difference of the square roots of each term of the

initial algebraic expression. The difference of squares is of the form

(ax2n – by2m), where a and b are squares of rational numbers, x and y are

variables, and n and m are natural numbers equal to or greater than 1 and

less than or equal to 4.

6. Factoring polynomials

Factor a polynomial containing up to six terms as the product of no

more than three prime factors by removing the common factor and

applying one other factoring method selected from the list below:

• factoring by grouping;

• factoring trinomials of the form x2 + bx + c or x2 + bxy + cy2;

• factoring trinomials of the form ax2 + bx + c or ax2 + bxy + cy2;

• factoring differences of squares.

The steps in the solution must be shown.

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7. Simplifying algebraic fractions

To simplify a rational algebraic fraction whose numerator and

denominator are factorable polynomials containing up to three

terms each. Each term contains no more than two variables. The

operation must be factored a maximum of four times, including no

more than two per polynomial. If a polynomial must be factored

twice, one of the factoring methods must involve removing the

common factor. The steps in the solution must be shown.

8. Multiplying and dividing two algebraic fractions

Find the product and quotient of two rational algebraic fractions.

The polynomials in the numerators and denominators are factorable

and contain at most three terms. Each term contains no more than

two variables. The solution must be factored a maximum of four

times, including no more than two per polynomial. If a polynomial

must be factored twice, one of the factoring methods must involve

removing the common factor. The product must be reduced to lowest

terms and the steps in the solution must be shown.

9. Adding and subtracting two algebraic fractions and comparing

algebraic expressions

Reduce to lowest terms an algebraic expression containing two

rational algebraic fractions joined by addition or subtraction. The

numerators and the denominators are factorable or non-factorable

polynomials, containing at most three terms. Each term contains no

more than two variables. If a polynomial must be factored twice, one

of the factoring methods must involve removing the common factor.

The common denominator must contain at most two binomials and

one monomial. The steps in the solution must be shown.

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Determine the equivalence of algebraic expressions by reducing

them to lowest terms. The expressions are made up by the sum or

difference of two algebraic fractions. The polynomials in the

numerators and denominators contain at most three terms. Each

term contains no more than two variables.

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DIAGNOSTIC TEST ON THE PREREQUISITES

Instructions

1. Answer as many questions as you can.

2. Do not use your calculator.

3. Write your answers on the test paper.

4. Do not waste any time. If you cannot answer a question, go on

to the next one immediately.

5. When you have answered as many questions as you can, correct

your answers using the answer key which follows the diagnostic

test.

6. To be considered correct, your answers must be identical to

those in the answer key. In addition, the various steps in your

solution should be equivalent to those shown in the answer key.

7. Copy your results onto the chart which follows the answer key.

This chart gives an analysis of the diagnostic test results.

8. Do the review activities that apply to your incorrect answers.

9. If all your answers are correct, you may begin working on this

module.

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1. Find all the factors of the following numbers.

a) 24: ............................................... b) 54: ..................................................

c) 100: .............................................

2. Reduce the following fractions to lowest terms.

a) 3575 = b) 12

64 =

c) 8127 =

3. Perform the following multiplications and divisions. Your results should be

reduced to lowest terms.

a) 814 × 7

12 = .................................. b) 1415 × 5

8 =.......................................

c) 611 ÷ 5

22 = .................................. d) 29 ÷ 2

3 = .........................................

4. Perform the following additions and subtractions. Your results should be

reduced to lowest terms.

a) 512 + 21

8 =...................................................................................................

b) 38 + 7

32 = .....................................................................................................

c) 187 – 2

3 = .....................................................................................................

d) 56 – 7

15 = .....................................................................................................

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5. Perform the following operations.

a) 4a2b + 6ab – 3a2b – a2b + 5ab = ..................................................................

b) (7yz + 2z – 3y) – (4z – 3yz + y) = .................................................................

c) 3cd(4d – 8c2 + cd2 – 2) = ..............................................................................

d) m

4n 2

3 – 2m2

5 + mn2

2 = .............................................................................

e) (2u + 3)(u – 4) = ...........................................................................................

f) (20p3q2 – 12p2q3 – 4p3q) ÷ 4pq =..................................................................

g) (3s + 4)2 = .....................................................................................................

h) 2r 2t2

3r2 + t

4 – 5rt7 ÷ – 3rt2

4 = ................................................................

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ANSWER KEY FOR THE DIAGNOSTIC TESTON THE PREREQUISITES

1. a) The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.

b) The factors of 54 are 1, 2, 3, 6, 9, 18, 27 and 54.

c) The factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50 and 100.

2. a) 3575 = 35 ÷ 5

75 ÷ 5 = 715 b) 12

64 = 12 ÷ 464 ÷ 4 = 3

16

c) 8127 = 81 ÷ 27

27 ÷ 27 = 31 = 3

3. a) 814 × 7

12 = 13 b) 14

15 × 58 = 7

3 × 14 = 7

12

c) 611 ÷ 5

22 = 611 × 22

5 = 61 × 2

5 = 125 d) 2

9 ÷ 23 = 2

9 × 32 = 1

3

4. a) 512 + 21

8 = 1024 + 63

24 = 7324 b) 3

8 + 732 = 12

32 + 732 = 19

32

c) 187 – 2

3 = 5421 – 14

21 = 4021 d) 5

6 – 715 = 25

30 – 1430 = 11

30

5. a) 4a2b + 6ab – 3a2b – a2b + 5ab = 11ab

b) (7yz + 2z – 3y) – (4z – 3yz + y) = 7yz + 2z – 3y – 4z + 3yz – y =

10yz – 4y – 2z

21

21

1

3

7

3

1

4

2

1

1

3

1

1

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c) 3cd(4d – 8c2 + cd2 – 2) = 12cd2 – 24c3d + 3c2d3 – 6cd

d) m

4n 2

3 – 2m2

5 + mn2

2 = mn2

12 – m3

10 + m2n 2

8

e) (2u + 3)(u – 4) = 2u2 – 8u + 3u – 12 = 2u2 – 5u – 12

f) (20p3q2 – 12p2q3 – 4p3q) ÷ 4pq = 5p2q – 3pq2 – p2

g) (3s + 4)2 = (3s + 4)(3s + 4) = 9s2 + 12s + 12s + 16 = 9s2 + 24s + 16

h) 2r 2t2

3r2 + t

4 – 5rt7 ÷ – 3rt2

4 = r 3t2

3 + r 2t3

6 – 10r 3t3

21 ÷ – 3rt2

4 =

r 3t2

3 + r 2t3

6 – 10r 3t3

21 × – 43rt2 = – 4r 2

9 – 2rt9 + 40r 2t

63

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ANALYSIS OF THE DIAGNOSTICTEST RESULTS

QuestionsAnswers Review Before going on to

Corrects Incorrects Section Page unit

1.a) 11.1 11.4 1b) 11.1 11.4 1c) 11.1 11.4 1

2.a) 11.2 11.9 7b) 11.2 11.9 7c) 11.2 11.9 7

3.a) 11.3 11.12 8b) 11.3 11.12 8c) 11.3 11.12 8d) 11.3 11.12 8

4.a) 11.4 11.17 9b) 11.4 11.17 9c) 11.4 11.17 9d) 11.4 11.17 9

5.a) 11.5 11.25 1b) 11.5 11.25 1c) 11.5 11.25 1d) 11.5 11.25 1e) 11.5 11.25 1f) 11.5 11.25 1g) 11.5 11.25 1h) 11.5 11.25 1

• If all your answers are correct, you may begin working on this module.

• For each incorrect answer, find the related section listed in the “Review”

column. Complete this section before beginning the unit listed in the right-

hand column under the heading “Before going on to unit”.

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INFORMATION FOR DISTANCEEDUCATION STUDENTS

You now have the learning material for MTH-4106-1 and the relevant homework

assignments. Enclosed with this package is a letter of introduction from your

tutor, indicating the various ways in which you can communicate with him or her

(e.g. by letter or telephone), as well as the times when he or she is available. Your

tutor will correct your work and help you with your studies. Do not hesitate to

make use of his or her services if you have any questions.

DEVELOPING EFFECTIVE STUDY HABITS

Learning by correspondence is a process which offers considerable flexibility, but

which also requires active involvement on your part. It demands regular study

and sustained effort. Efficient study habits will simplify your task. To ensure

effective and continuous progress in your studies, it is strongly recommended

that you:

• draw up a study timetable that takes your work habits into account and is

compatible with your leisure and other activities;

• develop a habit of regular and concentrated study.

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The following guidelines concerning theory, examples, exercises and

assignments are designed to help you succeed in this mathematics course.

Theory

To make sure you grasp the theoretical concepts thoroughly:

1. Read the lesson carefully and underline the important points.

2. Memorize the definitions, formulas and procedures used to solve a given

problem; this will make the lesson much easier to understand.

3. At the end of the assignment, make a note of any points that you do not

understand using the sheets provided for this purpose. Your tutor will then

be able to give you pertinent explanations.

4. Try to continue studying even if you run into a problem. However, if a major

difficulty hinders your progress, contact your tutor before handing in your

assignment, using the procedures outlined in the letter of introduction.

Examples

The examples given throughout the course are applications of the theory you are

studying. They illustrate the steps involved in doing the exercises. Carefully

study the solutions given in the examples and redo the examples yourself before

starting the exercises.

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Exercises

The exercises in each unit are generally modeled on the examples provided. Here

are a few suggestions to help you complete these exercises.

1. Write up your solutions, using the examples in the unit as models. It is

important not to refer to the answer key found on the coloured pages at the

back of the module until you have completed the exercises.

2. Compare your solutions with those in the answer key only after having done

all the exercises. Careful! Examine the steps in your solutions carefully,

even if your answers are correct.

3. If you find a mistake in your answer or solution, review the concepts that you

did not understand, as well as the pertinent examples. Then redo the

exercise.

4. Make sure you have successfully completed all the exercises in a unit before

moving on to the next one.

Homework Assignments

Module MTH-4106-1 comprises three homework assignments. The first page of

each assignment indicates the units to which the questions refer. The

assignments are designed to evaluate how well you have understood the

material studied. They also provide a means of communicating with your tutor.

When you have understood the material and have successfully completed the

pertinent exercises, do the corresponding assignment right away. Here are a few

suggestions:

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1. Do a rough draft first, and then, if necessary, revise your solutions before

writing out a clean copy of your answer.

2. Copy out your final answers or solutions in the blank spaces of the document

to be sent to your tutor. It is best to use a pencil.

3. Include a clear and detailed solution with the answer if the problem involves

several steps.

4. Mail only one homework assignment at a time. After correcting the

assignment, your tutor will return it to you.

In the section “Student’s Questions,” write any questions which you wish to have

answered by your tutor. He or she will give you advice and guide you in your

studies, if necessary.

In this course

Homework Assignment 1 is based on units 1 and 6.

Homework Assignment 2 is based on units 7 to 9.

Homework Assignment 3 is based on units 1 to 9.

CERTIFICATION

When you have completed all your work, and provided you have maintained an

average of at least 60%, you will be eligible to write the examination for this

course.

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UNIT 1

FACTORING BY REMOVING THECOMMON FACTOR

1.1 SETTING THE CONTEXT

It’s Common Knowledge!

Cindy and Jeff have just finished solving a mathematical problem. When they

compare results, they find that Jeff obtained the algebraic expression

4x5 + 12x4 + 8x3 for an answer, while Cindy got 4x3(x2 + 3x + 2).

At first glance, these results seem different. These two algebraic expressions

are, however, equivalent. In other words, they have the same value.

The expression 4x5 + 12x4 + 8x3 is a polynomial. Because this expression

contains three terms, it is called a trinomial.

START

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A polynomial is an algebraic expression made up of one or more

terms linked by plus and/or minus signs. If this expression

contains only one term, it is called a monomial . If it contains

two terms, it is a binomial; and if it is made up of three terms,

it is a trinomial.

What relationship can we find between the expressions 4x5 + 12x4 + 8x3 and

4x3(x2 + 3x + 2)? To answer this question, we have to factor a polynomial.

Factoring a polynomial means writing it in the form of the

product of two or more polynomials.

The algebraic expression 4x3(x2 + 3x + 2) that Cindy obtained is the product of the

monomial 4x3 and the trinomial x2 + 3x + 2. This expression is equivalent to

Jeff ’s answer because it is obtained by factoring the polynomial 4x5 + 12x4 + 8x3.

How this is done is what this unit is all about.

To reach the objective of this unit, you should be able to factor

polynomials of up to six terms by removing the common factor.

There are several methods of factoring. The method that can be used to solve

Cindy and Jeff ’s problem is called removing the common factor. To apply it,

we must first find the greatest common factor of all the terms of the polynomial

to be factored.

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A common factor, also known as a common divisor, is a number

or term that can divide several numbers or terms without a

remainder. For example, 5 is a common factor of 10 and 15

because 5 divides 10 and 15 without a remainder (10 ÷ 5 = 2 and

15 ÷ 5 = 3); 3x is a common factor of 6x and 9x2 because 3x divides

6x and 9x2 without a remainder 6x

3x = 2 et 9x 2

3x = 3x

Example 1

Find the greatest common factor of the binomial 3x2 + 6xy.

1. Find the greatest common factor of the numerical coefficients of both

terms of the binomial:

• the factors of 3 are 1, 3 ;

• the factors of 6 are 1, 2, 3 , 6.

∴ The greatest common factor of the numerical coefficients is 3.

The numerical coefficient is the number that multiplies the

variable or variables of a term. It is the numerical part of an

expression. Thus, the numerical coefficients of the expres-

sions 5x3; – 2y5; 13 ab 2 ; 0.5a3b5c4 are, respectively,

5; –2; 13 and 0.5.

2. Find the greatest common factor of the algebraic part of both terms of

the binomial. To do this:

a) find the variable or variables common to both terms:

• the variable x is common to both terms;

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b) assign each common variable the smallest exponent in the original

polynomial:

• in the binomial 3x2 + 6xy, 1 is the smallest exponent of the

variable x.

∴ The greatest common factor of the algebraic part is x.

3. Multiply each of the common elements: 3 × x = 3x.

∴ The greatest common factor of the binomial 3x2 + 6xy is 3x.

Example 2

Find the greatest common factor of the trinomial 10x4y3 + 4x3y – 2x2y2.

1. Find the greatest common factor of the numerical coefficients of all the

terms of the trinomial:

• the factors of 10 are 1, 2 , 5, 10;

• the factors of 4 are 1, 2 , 4;

• the factors of 2 are 1, 2 .

∴ The greatest common factor of the numerical coefficients of the

trinomial is 2.

2. Find the greatest common factor of the algebraic part of all the terms in

the trinomial. To do this:

a) find the variable or variables common to the three terms:

• the variables x and y are common to all three terms;

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b) assign each common variable the smallest exponent in the original

polynomial:

• in the trinomial 10x4y3 + 4x3y – 2x2y2, 2 is the smallest exponent of

the variable x and 1 is the smallest exponent of the variable y.

∴ The greatest common factor of the algebraic part is x2y.

3. Multiply each of the common elements: 2 × x2y = 2x2y.

∴ The greatest common factor of the expression 10x4y3 + 4x3y – 2x2y2 is

2x2y.

It’s as simple as that! Now let’s summarize the steps involved in finding the

greatest common factor of all the terms of a polynomial.

To find the greatest common factor of the terms of a

polynomial:

1. Find the greatest common factor of the numerical coeffi-

cients of all the terms of the polynomial.

2. Find the greatest common factor of the algebraic part of all

the terms of the polynomial by:

a) identifying the variable(s) common to all the terms of

the polynomial,

b) assigning each common variable the smallest expo-

nent in the original polynomial.

3. Multiply each of the common elements.

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? What is the greatest common factor of the expression

4a3b2c4 – 8a2b3c3 + 6b3c – 12a5b2c4?

1. .....................................................................................................................

2. .....................................................................................................................

3. .....................................................................................................................

If your answer was 2b2c, good for you! If not, read the following solution carefully

and redo the exercise.

1. • The factors of 4 are 1, 2 , 4;

• The factors of 8 are 1, 2 , 4, 8;

• The factors of 6 are 1, 2 , 3, 6;

• The factors of 12 are 1, 2 , 3, 4, 6, 12.

∴ The greatest common factor of the numerical coefficients is 2.

2. a) The variables common to all the terms of the polynomial are b and c;

b) 2 is the smallest exponent of the variable b and 1 is the smallest exponent

of the variable c.

∴ The greatest common factor of the algebraic part is b2c.

3. The greatest common factor of the algebraic expression

4a3b2c4 – 8a2b3c3 + 6b3c – 12a5b2c4 is 2 × b2c = 2b2c.

Let’s do some more exercises of this type. Knowing how to find the greatest

common factor of a polynomial is essential to understanding the upcoming

concepts.

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Exercise 1.1

Find the greatest common factor of the following polynomials.

1. 8x3 + 12x4 + 4x5

1. .....................................................................................................................

2. .....................................................................................................................

3. .....................................................................................................................

2. 6a2bc + 12a2b3c2 – 18a2b2

1. .....................................................................................................................

2. .....................................................................................................................

3. .....................................................................................................................

3. 16x3y + 12x3y2 – 8x2y4 + 20x2y2

1. .....................................................................................................................

2. .....................................................................................................................

3. .....................................................................................................................

4. 2m2np – 3mn2p2 + 5m3n3p

1. .....................................................................................................................

2. .....................................................................................................................

3. .....................................................................................................................

5. 3k3l4 – 6k2l3 + 18k4l2 – 12k5l3 + 3k3l2

1. .....................................................................................................................

2. .....................................................................................................................

3. .....................................................................................................................

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Having completed these mental gymnastics, you can now solve Cindy and Jeff ’s

problem, using the method of factoring by removing the common factor. The

example below shows you the procedure to follow.

Example 3

Factor the trinomial 4x5 + 12x4 + 8x3 obtained by Jeff.

1. Find the greatest common factor of all the terms of the trinomial:

• the greatest common factor of the numerical coefficients of all the

terms is 4;

• the greatest common factor of the algebraic part of all the terms is x3.

∴ The greatest common factor of the trinomial 4x5 + 12x4 + 8x3 is 4x3 .

2. Divide each term of the trinomial by this common factor.

4x5

4x3 + 12x4

4x3 + 8x3

4x3 = x5 – 3 + 3x4 – 3 + 2x3 – 3 = x2 + 3x + 2.

• To perform the division of two monomials, divide the

numerical coefficients of the two monomials and subtract

the exponents of the same variable:

am

an = am – n.

• Any variable raised to the power of 0 is equal to 1: a0 = 1.

3. Put the new trinomial in parentheses and write the greatest common

factor in front of the parentheses. We thereby remove the common

factor: 4x3(x2 + 3x + 2).

1

1

3

1

2

1

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4. Check the answer.

To do this, multiply each of the terms in parentheses by the monomial in

front of the parentheses.

4x3(x2 + 3x + 2) = 4x3(x2) + 4x3(3x) + 4x3(2) = 4x3 + 2 + 12x3 + 1 + 8x3 =

4x5 + 12x4 + 8x3

Since the polynomial obtained is equal to the original polynomial, the

polynomial has been factored. Thus:

4x3(x2 + 3x + 2) = 4x5 + 12x4 + 8x3

To multiply a monomial by a polynomial, multiply the

numerical coefficient of the monomial by each numerical

coefficient in the polynomial and add the exponents of the

same variable: am × an = am + n.

☞ Checking your answer is an important step, since it allows you to

make sure that you performed the division in Step 2 correctly. It also

allows you to identify incorrect signs or an incorrect greatest

common factor. It does not, however, allow you to determine with

certainty that you correctly identified the greatest common factor.

For example, if in Example 3, you found 4x2 in Step 1, you will obtain

4x2(x3 + 3x2 + 2x) = 4x5 + 12x4 + 8x3, which appears to be correct.

However, the initial expression has not been factored completely

since, in the parentheses, the factor x is still common to the three

terms of the polynomial. It is therefore crucial to make sure that you

correctly determined the greatest common factor of all the terms

of the polynomial from which you want to remove the common factor.

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Jeff and Cindy were therefore both right. Cindy arrived at her solution by

factoring the expression that Jeff obtained. Let’s go on to another example.

Example 4

Factor the following polynomial by removing the common factor:

–3m3n – 7m3r + 8m3rt

1. Find the greatest common factor of all the terms of the polynomial:

• the greatest common factor of the numerical coefficients of all the

terms is 1;

• the greatest common factor of the algebraic part is m3.

∴ The greatest common factor of the polynomial is m3.

2. Divide each term of the polynomial by this common factor.

–3m3nm3 – 7m3r

m3 + 8m3rtm3 = –3m3 – 3n – 7m3 – 3r + 8m3 – 3rt

= –3n – 7r + 8rt

3. Put the new polynomial in parentheses and write the greatest common

factor in front of the parentheses.

m3 (–3n – 7r + 8rt)

4. Check the answer:

m3(–3n – 7r + 8rt ) = –3m3n – 7m3r + 8m3rt

Since the product is equal to the original polynomial, the polynomial has

been factored. Thus:

–3m3n – 7m3r + 8m3rt = m3 (–3n – 7r + 8rt)

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N.B. In the previous example, we can also remove the factor –m3. In such a case,

it is important to pay attention to the signs joining each term of the trinomial in

parentheses. After factoring, we obtain –m3(3n + 7r – 8rt) because:

–m3(3n + 7r – 8rt) = –3m3n – 7m3r + 8m3rt

Law of signs for multiplication or division

+ times + = +

– times – = +

+ times – = –

– times + = –

Factoring by removing the common factor is a snap! Let’s summarize the steps

involved in applying this method.

To factor a polynomial by removing the common

factor:

1. Find the greatest common factor of all the terms of the

polynomial.

2. Divide each term of the polynomial by this common factor.

3. Put the new polynomial in parentheses and write the

greatest common factor in front of the parentheses.

4. Check the answer by multiplying the isolated factor by

each of the terms of the polynomial in parentheses.

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? Factor the polynomial –2ab3 – 4b3c – 12b3d by removing the common factor.

1. .....................................................................................................................

2. .....................................................................................................................

3. .....................................................................................................................

4. .....................................................................................................................

To factor this polynomial, you must remove the common factor –2b3 or the

common factor 2b3.

1. The greatest common factor of all the terms of the polynomial is –2b3 or 2b3.

2. –2ab3

–2b3 – 4b3c–2b3 – 12b3d

–2b3 = a + 2c + 6d

or

– 2ab3

2b3 – 4b3c2b3 – 12b3d

2b3 = –a – 2c – 6d

3. –2b3(a + 2c + 6d)

or

2b3(–a – 2c – 6d)

4. –2b3(a + 2c + 6d) = –2ab3 – 4b3c – 12b3d

or

2b3(–a – 2c – 6d) = –2ab3 – 4b3c – 12b3d

1

1

2

1

6

1

1

1

2

1

6

1

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Since the product is equal to the original polynomial, the polynomial has been

factored completely. Thus:

–2b3(a + 2c + 6d) = –2ab3 – 4b3c – 12b3d

or

2b3(–a – 2c – 6d) = –2ab3 – 4b3c – 12b3d

N.B. If the first term of the polynomial to be factored is negative, it is preferable

to remove a common factor with a negative sign.

To become an ace at factoring, nothing beats practice! The following exercises

will help you improve your skills.

Exercise 1.2

Factor the following polynomials by removing the common factor, following the

steps described above.

1. a3 – ax

1. .....................................................................................................................

2. .....................................................................................................................

3. .....................................................................................................................

4. .....................................................................................................................

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2. 5ab – 5a3b2

1. .....................................................................................................................

2. .....................................................................................................................

3. .....................................................................................................................

4. .....................................................................................................................

3. a2bc + ab2c + abc2

1. .....................................................................................................................

2. .....................................................................................................................

3. .....................................................................................................................

4. .....................................................................................................................

4. 18a3 – 24a3b + 12ab2

1. .....................................................................................................................

2. .....................................................................................................................

3. .....................................................................................................................

4. .....................................................................................................................

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5. 12x3y2 – 8x2y3 – 4xy

1. .....................................................................................................................

2. .....................................................................................................................

3. .....................................................................................................................

4. .....................................................................................................................

6. 8m3n3 – 12m2n2p5 + 20m5np4

1. .....................................................................................................................

2. .....................................................................................................................

3. .....................................................................................................................

4. .....................................................................................................................

7. 11a2x3y – 22b2x2y2 + 33c2x2yz

1. .....................................................................................................................

2. .....................................................................................................................

3. .....................................................................................................................

4. .....................................................................................................................

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8. 12a2x3y2 + 9ax2y3 – 15a4x4y5

1. ....................................................................................................................

2. ....................................................................................................................

3. ....................................................................................................................

4. ....................................................................................................................

9. –h2k – k2 – k3

1. ....................................................................................................................

2. ....................................................................................................................

3. ....................................................................................................................

4. ....................................................................................................................

10. –7r2st + 14rs2t2u – 21r2s2t2 – 7r3s2t

1. ....................................................................................................................

2. ....................................................................................................................

3. ....................................................................................................................

4. ....................................................................................................................

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To factor a polynomial by removing the common factor, you simply have to know

how to find the greatest common factor of a polynomial and to divide two

monomials. But watch the signs when you remove a negative factor!

Checking your answer becomes very important in this instance, for it allows you

to make sure that the sign of each term obtained after multiplying is the same

as in the original polynomial.

Now before going on to the practice exercises, let’s have some fun!

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Did you know that...

… in the following addition exercise, if you replace each of

the letters A, B, C, D, E, F, G with a number from 1 to 7, you

can get the sum of 9 999 999?

Careful! Each letter corresponds to a specific number.

2 F C 8 E E 0

D 5 9 D 4 9 A

G D 6 1 A E G

+ B A 7 C G C 3

9 9 9 9 9 9 9

Solution

A = 4, B = 1, C = 6, D = 3, E = 7, F = 5 et G = 2

256877035934942361472

+14762639999999

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1.2 PRACTICE EXERCISES

Factor the following polynomials by removing the common factor.

1. x2y3 – x2y2 + x2y =

2. 10x3 – 25x4y =

3. –16c + 64c2d =

4. 6a2b3 + 14a4b3c =

5. 38x3y5 + 57x4y2 =

?

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6. 12 m2n + 1

4 mn2 =

7. 5ax5 – 10a2x3 – 15a3x3 =

8. 48a3b2c + 24a3bc3 – 16ab3c3 + 32ab2c4 =

9. 3a4b2 – 3a3b + 6a2b – 9ab3 + 9a3b2 – 12a2b =

10. 5.2m3n2 + 10.4m3n3 + 15.6m2n3 =

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11. 12mx4y3 – 18nx3y – 21x2y4 + 6x2yz =

12. –9b2 – 81b =

13. –8x3yz3 – 12x2y3z5 + 20x5yz4 =

14. 15x3y – 12x4y3z + 7xy2 =

15. –8m2n3 – 4m4n2 – 16m3n2 =

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1.3 REVIEW EXERCISES

1. Complete the following sentences by filling in the blanks with the missing

term or expression.

To find the greatest common factor of the terms of a polynomial, find the

greatest common factor of the ................................... .................................... of

all the ..................................... of the polynomial. Then find the greatest

common factor of the ................................................. part of all the terms of

the polynomial. To do this, identify the ..................................................(s)

common to all the terms of the polynomial and assign each of them the

............................................ exponent in the original polynomial. Lastly,

multiply each of the ......................................... elements.

2. Explain in your own words what factoring a polynomial means.

...........................................................................................................................

...........................................................................................................................

3. List the four steps in factoring a polynomial by removing the common factor.

...........................................................................................................................

...........................................................................................................................

...........................................................................................................................

...........................................................................................................................

...........................................................................................................................

...........................................................................................................................

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1.4 THE MATH WHIZ PAGE

Uncommon Common Factors

The greatest common factor of a polynomial may be a binomial. For

example, in the expression 3x(2a + b) – 5y(2a + b), the binomial

(2a + b) is the common factor of both terms of the algebraic expression.

To factor this type of expression, you need only follow the procedure

shown in this unit. Thus, by dividing each term of the algebraic

expression by the common factor, we get:

3x(2a + b)(2a + b)

– 5y(2a + b)(2a + b)

= 3x – 5y

You then simply put the new polynomial in parentheses and write the

common factor in front of the parentheses. The result of the factoring

is therefore:

(2a + b)(3x – 5y)

Following the same reasoning, factor the algebraic expressions shown

below. Careful! In some cases, one of the two terms obtained may need

to be factored a second time.

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1. y(y – 1) + 2(y – 1) =

2. (a – 5)a – 3(a – 5) =

3. 5x(a + b) + 15y(a + b) – 10z(a + b) =

4. 6x2(3a – b) – 15x(3a – b) =

Documents

MTH-4106-1 F actoring - SOFAD · PDF file3. factoring trinomials of the form ax2 + bx + c or ax2 + bxy + cy2; 5. factoring differences of squares. To factor a polynomial is to write