mth-4110-1 the f operations to be met. i see!? start lastly, the finish line indicates ......

$: MTH-4110-1 The F Operations to be met. I see!? START Lastly, the finish line indicates ... Simplifying Algebraic Fractions Reduce a rational algebraic fraction to its lowest terms$
The Four

Operations

on

Algebraic Fractions

MTH-4110-1

MTH-4110-1 C1 10/5/11 12:01 PM Page 1

THE FOUR

OPERATIONS

ON

ALGEBRAIC

FRACTIONS

MTH-4110-1

Author: Suzie Asselin

Content revision: Daniel GélineauJean-Paul GroleauMireille Moisan-SanscartierNicole Perreault

Adult Education Consultants: Les Productions C.G.L. enr.

Coordinator for the DDFD: Jean-Paul Groleau

Coordinator for the DFGA: Ronald Côté

Word processing: Francine Lessard

Photocomposition and layout: Multitexte Plus

English version: Direction du développement pédagogique en langueanglaise

Translation: Elizabeth Dundas

Linguistic revision: William Gore

Translation of updated sections: Claudia de Fulviis

Reprint: 2004

© Société de formation à distance des commissions scolaires du Québec

All rights for translation and adaptation, in whole or in part, reserved for all countries.

Any reproduction by mechanical or electronic means, including micro-reproduction, is

forbidden without the written permission of a duly authorized representative of the

Société de formation à distance des commissions scolaires du Québec (SOFAD).

Legal Deposit — 2004Bibliothèque et Archives nationales du QuébecBibliothèque et Archives CanadaISBN 2-89493-288-9

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MTH-4110-1 The Four Operations on 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 ....................................................... 0.13Answer Key for the Diagnostic Test on the Prerequisites ...................... 0.17Analysis of Diagnostic Test Results ......................................................... 0.19Information for Distance Education Students......................................... 0.21

UNITS

1. Simplifying Algebraic Fractions ............................................................... 1.12. Product and Quotient of Algebraic Fractions .......................................... 2.13. Multiplying and Dividing Algebraic Fractions ........................................ 3.14. Adding and Subtracting Algebraic Fractions .......................................... 4.15. Order of Operations Involving Algebraic Fractions ................................ 5.1

Final Summary.......................................................................................... 6.1Answer Key for the Final Summary ........................................................ 6.5Terminal Objective .................................................................................... 6.6Self-Evaluation Test.................................................................................. 6.7Answer Key for the Self-Evaluation Test ................................................ 6.13Analysis of the Self-Evaluation Test Results .......................................... 6.17Final Evaluation........................................................................................ 6.18Answer Key for the Exercises ................................................................... 6.19Glossary ..................................................................................................... 6.41List of Symbols .......................................................................................... 6.45Bibliography .............................................................................................. 6.46

Review Activities ....................................................................................... 7.1

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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 Gegep-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-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

MAT-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

THE 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 education

center or atFormation àdistance, ...

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 exerciceswhich 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.

I see!

?

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 a bighelp.

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

ONE STEP FURTHER IN MATHEMATICS

WITH ALGEBRAIC FRACTIONS

If you are continuing your studies in mathematics or you are taking science

courses, you will have to deal with mathematical expressions containing one or

more variables. Some expressions will be in the form of a fraction whose

numerator or denominator is a monomial or a polynomial. Such algebraic

expressions are known as algebraic fractions. For example, the expressions

x 2 + 6x + 82x + 4 , 4ab2

9 , 19m2 – 4n 2 and p2q2

2p2q2 – 5pq2 – 3q2

are algebraic fractions.

In this module, you will learn how to perform various operations on algebraic

fractions. You will first learn how to simplify them. It is important to master

this skill before going on, for you will use it throughout the module. Indeed, all

your results will have to be reduced to lowest terms.

In the following units, you will learn how to multiply and divide algebraic

fractions, simplify algebraic expressions involving the multiplication and divi-

sion of algebraic fractions, add and subtract algebraic fractions and finally,

simplify algebraic expressions that may involve the four operations on algebraic

fractions. In this last case, you need to apply the rules for the order of

operations, which you probably already know.

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

help you a great deal here. You will also find it useful to review the five methods

of factoring polynomials, also known as finding the factors of a polynomial:

• factoring by removing the common factor;

• 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.

These are the main concepts that will be used in this module on algebraic

fractions.

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INTERMEDIATE AND TERMINAL OBJECTIVES OFTHE MODULE

Module MTH-4110-1 consists of five units and requires 25 hours of study

distributed as shown below. Each unit covers either an intermediate or a

terminal objective. The terminal objective appears in boldface.

Objectives Number of Hours* % (evaluation)

1 to 5 24 100%

* One hour are allotted for the final evaluation.

1. Simplifying Algebraic Fractions

Reduce a rational algebraic fraction to its lowest terms. The numerator and

denominator are factorable polynomials that contain up to three terms each

and each term contains no more than two variables. The steps involved in

simplifying the fraction must be shown.

2. Multiplying and Dividing Algebraic Fractions

Multiply three rational algebraic fractions and divide two rational algebraic

fractions. The polynomials in the numerators and denominators are

factorable and contain up to three terms. Each term contains no more than

two variables. The product and quotient must be reduced to their lowest

terms and the steps in the solution must be shown.

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3. Simplifying Algebraic Expressions Containing Algebraic Fractions That Are

Multiplied and Divided

Simplify an algebraic expression containing up to four rational algebraic

fractions that are multiplied and divided. The numerators and denominators

are factorable polynomials that contain up to three terms each and each term

contains no more than two variables. The steps involved in simplifying the

expression must be shown.

4. Simplifying Algebraic Expressions Containing Algebraic Fractions That Are

Added and Subtracted

Simplify an algebraic expression containing up to three rational algebraic

fractions that are added and subtracted. The numerators and denominators

are factorable polynomials that contain up to three terms each and each term

contains no more than two variables. The steps involved in simplifying the

expression must be shown.

5. Order of Operations Involving Algebraic Fractions

Simplify an algebraic expression containing up to three rational

algebraic fractions by performing the appropriate operations and

by following the order of operations. The algebraic expression

contains no more than two sets of parentheses. The numerators and

denominators are factorable polynomials that contain up to three

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

steps involved in simplifying the expression must be shown.

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DIAGNOSTIC TEST ON THE PREREQUISITES

Instructions

1. Answer as many questions as you can.

2. You may not use a calculator.

3. Write your answers on the test paper.

4. Do not waste any time. If you cannot answer a question, go onto the next one immediately.

5. When you have answered as many questions as you can, correctyour answers using the answer key which follows the diagnostictest.

6. To be considered correct, answers must be identical to those inthe key. In addition, the various steps in your answer should beequivalent to those shown in the solution.

7. Transcribe your results onto the chart which follows the answerkey. It gives an analysis of the diagnostic test results.

8. Do only the review activities which are suggested for each ofyour incorrect answers.

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

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1. Reduce each fraction to lowest terms.

a) 6080 =

..........................................b) 26

65 =...........................................

2. Factor the following polynomials.

a) 4a2 + 8ab = ...................................................................................................

b) x2 + 2x – 3 = .................................................................................................

c) h2 – 25k2 = ....................................................................................................

d) uw + 2vw – 3uv – 6v2 = ...............................................................................

e) 4 – 9j 2 = .......................................................................................................

f) 2z2 – 13z – 7 = ..............................................................................................

g) –d2 – d + 2 = ................................................................................................

h) 4m2 – 5mn + n2 = .........................................................................................

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

i) –r2 + 14rs – 49s2 = .......................................................................................

j) –36p2 + 4q2 = ...............................................................................................

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

reduced to lowest terms.

a) 910 × 2

45 =....................................................................................................

b) 37 × 14

15 × 58 =

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

c) 34 ÷ 7

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

d) 29 ÷ 2

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

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

reduced to lowest terms.

a) 38 + 7

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

b) 25 + 3

7 =........................................................................................................

c) 56 – 7

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

d) 113 – 1

10 =....................................................................................................

5. Perform the following operations.

a) (c2 + 3cd) + (2c2 – 5cd) = ..............................................................................

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

b) (t2 – 7t + 2) – (t2 + 7t – 10) = ........................................................................

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

c) (2d2 + 3dg) + (g2 – 3g2) – (d2 – g2 – 2d2) =

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

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

d) 3y(2y2 + 4xy + 2x2) = ....................................................................................

e) (a + 2b)(3a – b) = .........................................................................................

f) 13 x2 – 1

2

2

=................................................................................................

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g) 25a3b2

5ab+ (3a2 + 2b)2b =

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

h)

12 xy 1

3 x2y – 12 xy + 2

3 x2y ÷ 23 xy =

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

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

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

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ANSWER KEY FOR THE DIAGNOSTIC TESTON THE PREREQUISITES

1. a) 6080 = 3 × 20

4 × 20 = 34 b) 26

65 = 2 × 135 × 13 = 2

5

2. a) 4a2 + 8ab = 4a(a + 2b)

b) x2 + 2x – 3 = (x + 3)(x – 1)

c) h2 – 25k2 = (h + 5k)(h – 5k)

d) uw + 2vw – 3uv – 6v2 = w(u + 2v) – 3v(u + 2v) = (u + 2v)(w – 3v)

e) 4 – 9j2 = (2 + 3j)(2 – 3j)

f) 2z2 – 13z – 7 = 2z2 – 14z + z – 7 = 2z(z – 7) + 1(z – 7) = (z – 7)(2z + 1)

g) –d2 – d + 2 = –(d2 + d – 2) = –(d + 2)(d – 1) or (d + 2)(1 – d)

h) 4m2 – 5mn + n2 = 4m2 – 4mn – mn + n2 = 4m(m – n) – n(m – n) =

(m – n)(4m – n)

i) –r2 + 14rs – 49s2 = –(r2 – 14rs + 49s2) = –(r – 7s)2

j) –36p2 + 4q2 = –4(9p2 – q2) = –4(3p + q)(3p – q)

3. a) 910 × 2

45 = 1 × 15 × 5 = 1

25

b) 37 × 14

15 × 58 = 1 × 1 × 1

1 × 1 × 4 = 14

c) 34 ÷ 7

8 = 34 × 8

7 = 3 × 21 × 7 = 6

7

d) 29 ÷ 2

3 = 29 × 3

2 = 1 × 13 × 1 = 1

3

1 2 1

11 3 4

1

1 1

5 5

2

1

1

3

1

1

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4. a) 38 + 7

32 = 1232 + 7

32 = 1932 b) 2

5 + 37 = 14

35 + 1535 = 29

35

c) 56 – 7

15 = 2530 – 14

30 = 1130 d) 1

13 – 110 = 10

130 – 13130 = –3

130

5. a) (c2 + 3cd) + (2c2 – 5cd) = c2 + 2c2 + 3cd – 5cd = 3c2 – 2cd or c(3c – 2d)

b) (t2 – 7t + 2) – (t2 + 7t – 10) = t2 – 7t + 2 – t2 – 7t + 10 = –14t + 12 or –2(7t – 6)

c) (2d2 + 3dg) + (g2 – 3g2) – (d2 – g2 – 2d2) =

2d2 + 3dg + g2 – 3g2 – d2 + g2 + 2d2 = 3d2 + 3dg – g2

d) 3y(2y2 + 4xy + 2x2) = 6y3 + 12xy2 + 6x2y

e) (a + 2b)(3a – b) = 3a2 + 6ab – ab – 2b2 = 3a2 + 5ab – 2b2

f) 13 x2 – 1

2

2

= 13 x2

2

– 2 13 x2 × 1

2 + –12

2

= 19 x4 – 1

3 x2 + 14

g) 25a3b2

5ab+ (3a2 + 2b)2b = 5a2b + 6a2b + 4b2 = 11a2b + 4b2

h) 12 xy 1

3 x 2y – 12 xy + 2

3 x 2y ÷ 23 xy = 1

6 x 3y 2 – 14 x 2y 2 + 1

3 x 3y 2 ÷ 2xy3 =

x 3y 2

2 – x 2y 2

4 × 32xy = 3x 2y

4 – 3xy8 or 3

4 x 2y – 38 xy

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ANALYSIS OF THE DIAGNOSTICTEST RESULTS

QuestionAnswer Review Before Going

Correct Incorrect Section Page to Unit(s)

1. a) 7.1 7.4 1 to 5b) 7.1 4.4 1 to 5

2. a) 7.2 7.7 1 to 5b) 7.2 7.7 1 to 5c) 7.2 7.7 1 to 5d) 7.2 7.7 1 to 5e) 7.2 7.7 1 to 5f) 7.2 7.7 1 to 5g) 7.2 7.7 1 to 5h) 7.2 7.7 1 to 5i) 7.2 7.7 1 to 5j) 7.2 7.7 1 to 5

3. a) 7.3 7.23 2, 3 and 5b) 7.3 7.23 2, 3 and 5c) 7.3 7.23 2, 3 and 5d) 7.3 7.23 2, 3 and 5

4. a) 7.4 7.28 4 and 5b) 7.4 7.28 4 and 5c) 7.4 7.28 4 and 5d) 7.4 7.28 4 and 5

5. a) 7.5 7.37 3 to 5b) 7.5 7.37 3 to 5c) 7.5 7.37 3 to 5d) 7.5 7.37 3 to 5e) 7.5 7.37 3 to 5f) 7.5 7.37 3 to 5g) 7.5 7.37 3 to 5h) 7.5 7.37 3 to 5

• 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. Do the review activities for that section before beginning the units

listed in the right-hand column under the heading Before Going to Unit(s).

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INFORMATION FOR DISTANCEEDUCATION STUDENTS

You now have the learning material for MTH-4110-1 together with the home-

work assignments. Enclosed with this material 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, 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

Distance education 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 working habits into account and

is compatible with your leisure time and other activities;

• develop a habit of regular and concentrated study.

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The following guidelines concerning the theory, examples, exercises and assign-

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

Theory

To make sure you thoroughly grasp the theoretical concepts:

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

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

problem, since this will make the lesson much easier to understand.

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

understand. Your tutor will then be able to give you pertinent explanations.

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

if a major difficulty hinders your learning, ask for explanations before

sending in your assignment. Contact your tutor, using the procedure

outlined in his or her letter of introduction.

Examples

The examples given throughout the course are an application 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 them yourself before starting

the exercises.

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Exercises

The exercises in each unit are generally modelled 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

end 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 solution carefully even

if your answers are correct.

3. If you find a mistake in your answer or your 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-4110-1 contains three 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 done the perti-

nent exercises, do the corresponding assignment immediately. Here are a few

suggestions.

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

submitting a clean copy of your answer.

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

to be sent to your tutor. It is preferable 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 assign-

ment, your tutor will return it to you.

In the section “Student’s Questions”, write any questions which you may 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 to 4.

Homework Assignment 2 is based on unit 5.

Homework Assignment 3 is based on units 1 to 5.

CERTIFICATION

When you have completed all the 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

SIMPLIFYING ALGEBRAIC FRACTIONS

1.1 SETTING THE CONTEXT

Special Fractions

So you've long been an ace at simplifying fractions! In the wink of an eye, you

can reduce the following fractions to lowest terms.

Prove it to yourself by filling in the blanks below.

? The simplest expression of 48 is ............. .

? If we simplify the fraction 570 , we get ............... .

? By reducing 13143 to lowest terms, we obtain ............... .

? After simplifcation, the fraction 18171 becomes ............... .

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13143 = 1

1118171 = 1

19

48 = 1

25

70 = 114

You probably obtained the following answers: 12 , 1

14 , 111 and 2

19 .

To arrive at these answers, you had to

find the greatest common factor of the

numerator and of the denominator.

• The common factor of a fraction is a number by which both

terms can be divided.

• In any fraction of the form ab

, the term a is called the

numerator and the term b is called the denominator.

Do you know how to simplify algebraic fractions like these ones?

4x8x 2 , 42ab2c3

3a2b, 2m + 8

m2 + 6m + 8

Simplifying algebraic fractions will be very useful should you decide to continue

studying mathematics or science. This new skill will enable you to solve various

problems in trigonometry, geometry, differential and integral calculus and other

fields.

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

algebraic fractions to lowest terms.

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Algebraic fractions are fractions whose numerator and

denominator are monomials or polynomials.

• A monomial is an algebraic expression consisting of a

single term which can be a number, a variable or a product

of numbers and variables.

E.g. 2m, 42ab2c3, 8, m2.

• A polynomial is an algebraic expression made up of a term or

a group of terms, that are joined by addition or subtraction

signs.

E.g. 5x2, 2m + 8, 36a2b3c2 + 6abc + bc2.

To reduce an algebraic fraction to lowest terms, simply apply the method used

for numerical fractions, which consists in dividing the numerator and the

denominator by the greatest common factor.

Example 1

Reduce the following algebraic fractions to lowest terms:

4x8x 2 , 42ab2c3

3a2b, 2m + 8

m2 + 6m + 8

a) 4x8x 2 = 1 × 4x

2x × 4x = 12x

• The algebraic expression 4x is the greatest common factor of both the

numerator and the denominator.

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b) 42ab2c3

3a2b= 14bc3 × 3ab

a × 3ab= 14bc3

a

• 3ab is the greatest common factor of the numerator and the denomi-

nator.

c) 2m + 8m2 + 6m + 8

= 2 × (m + 4)(m + 2) × (m + 4)

= 2m + 2

• (m + 4) is the greatest common factor of the numerator and the

denominator.

To simplify algebraic fractions, it is necessary to factor the numerator and the

denominator.

Factoring an algebraic expression means breaking it down into

the product of prime factors, that is, factors that cannot them-

selves be broken down into factors.

The five methods of factoring are:

• factoring by removing the common factor

e.g. m2 + 3m = m(m + 3);

• factoring by grouping

e.g. p2 + 2pq + pr + 2qr = (p + 2q)(p + r);

• factoring a trinomial of the form x2 + bx + c

or of the form x2 + bxy + cy2

e.g. a2 + 7ab + 12b2 = (a + 3b)(a + 4b);

• factoring a trinomial of the form ax2 + bx + c

or of the form ax2 + bxy + cy2

e.g. 2z2 – 9z – 5 = (2z + 1)(z – 5);

• factoring differences of squares

e.g. 4x2 – 9y2 = (2x – 3y)(2x + 3y).

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To reduce an algebraic fraction to lowest terms:

1. Factor the numerator and the denominator if possible.

2. Simplify the fraction by dividing the numerator and the

denominator by the common factors.

Take a close look at Example 2 before going on to the exercises.

Example 2

a) Simplify the algebraic fraction p2 – 42p2 + 7p + 6

.

1. Factor the numerator and the denominator:

p2 – 4 = (p – 2)(p + 2) (difference of squares)

2p2 + 7p + 6 = (2p + 3)(p + 2) (trinomial of the form x2 + bx + c)

The algebraic fraction becomes: (p – 2)(p + 2)(2p + 3)(p + 2)

2. The common factor of the numerator and the denominator is (p + 2).

Remove this factor:

(p – 2)(p + 2)(2p + 3)(p + 2)

= p – 22p + 3

This is the simplest expression of the fraction p2 – 42p2 + 7p + 6

.

b) Simplify the algebraic fraction 2x – 143x – 21 + bx – 7b

.

2x – 143x – 21 + bx – 7b

= 2(x – 7)(x – 7)(3 + b)

= 23 + b

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N.B. In this case, two methods of factoring are used: removing the

common factor in the numerator and factoring by grouping in the denomi-

nator.

Now it's your turn to practise this method in the following exercise!

Exercise 1.1

Reduce each of the following algebraic fractions to lowest terms.

1. 6g9g =

2. 15s2tu5

27stu 2 =

3. a2 – a2bc3 – c3b

=

4. 2x + 6x 2 + 5x + 6

=

5. v 2 – v – 12v – 4 =

6. 6m – 128m – 16 =

7. ab + bc + a2 + acb2 – a2

=

8. r 2 – 2r – 154r 2 + 13r + 3

=

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N.B. If you had difficulty solving the preceding problems, reread the explana-

tions and examples from the beginning of this unit. Practise factoring

polynomials in particular, as you will have to do this throughout this module.

Don't hesitate to do the review activities or consult a resource person if

necessary.

Notes

Simplifying algebraic fractions is not always that simple, and you should beware

of a number of traps.

1. Only identical factors found in the numerator and the denominator of the

fraction can be eliminated. Identical terms in the polynomials that make up

an algebraic fraction cannot be removed. For example, in the algebraic

fraction a2 + b2

b2 , the term b2 cannot be cancelled out, as b2 is not a factor of

the polynomial a2 + b2. In other words, a2 + b2

b2≠ a2 .

2. The factor (a + b) is identical to the factor (b + a), since the order in which

the terms are placed does not matter. Thus (a + b) = (b + a).

3. The factors (a – b) et (b – a) are not identical! We can, however, transform

one of these factors. Thus,

(b – a) = + b – a = –1(–b + a) = – (a – b).

We can ensure that the law of signs is properly applied by performing the

inverse operation: – (a – b) = –a + b = b – a.

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Law of signs for multiplication and division:

+ × + = + + ÷ + = +

– × – = + and – ÷ – = +

+ × – = – + ÷ – = –

– × + = – – ÷ + = –

Example 3

Simplify the fraction y 2 – 1y – y 2 .

1. Factor the numerator and the denominator.

(y + 1)(y – 1)y(1 – y)

= (y + 1)(y – 1)– y(–1 + y)

= (y + 1)(y – 1)– y(y – 1)

2. Simplify by removing the common factor.

(y + 1)(y – 1)– y(y – 1)

The answer is y + 1– y or – (y + 1)

y or 1 – yy .

N.B. Certain algebraic fractions cannot be simplified. To determine which

ones cannot, it is still necessary to factor the numerator and the denominator.

Example 4

The algebraic fraction r 2 + 8rs + 12s2

r 2 + 7rs + 12s2 cannot be simplified because, after

factoring, we obtain the fraction (r + 6s)(r + 2s)(r + 4s)(r + 3s)

, which does not contain any

common factor in the numerator and the denominator. This fraction is

therefore said to be irreducible.

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Exercise 1.2

Reduce the following algebraic fractions to lowest terms.

1. h 2 – h 2k2

h 3 – h 3k2 =

2. ab – bc– b

=

3. 4 – jj 2 – 16

=

4. 4x 2 – 8xy – 12y 2

3y – x =

5. 21m – 3nn 2 – 49m2 =

6. 2d – 62d 2 – 7d + 6

=

N.B. Before doing the practice exercises, make sure that you have understood the

exercises in this series. It is essential that you master the objective of this unit,

for you will have to apply it in all of the following units.

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Did you know that...

sometimes a little logic can be a life saver?

Suppose you were in the following rather unfortunate

situation: you are being held by an over-zealous executioner who wants to

test his wits against yours. To make the situation even more interesting,

he is willing to let you choose your punishment. The rules are:

• if you tell the truth, you will be hanged;

• if you tell a lie, you will be beheaded.

The executioner knows, however, that you

can say one sentence that would make it

impossible for him to execute you. Hurry up

and find that sentence!

Solution:

By now, you should be ready to tackle the practice exercises!

The sentence is, “I will be beheaded.”•If it is the truth, you must be hanged. But if you are hanged, what you have just said

is a lie, and the punishment is wrong.•If it is a lie, you will be beheaded. But if you are beheaded, what you have just said

is the truth, and the punishment is again wrong.In both cases, the executioner cannot execute you... Whew!

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?1.2 PRACTICE EXERCISES

Reduce the algebraic fractions below to lowest terms.

1. 6x 2 – 8xy9xy – 12y 2 =

2. 15a2b3

– 3a3b =

3. 2m + 2n(m + n)2 =

4. c2 – 4d 2

2d + c =

5. (2 j + 6)2

4 j 2 – 36 =

6. 3p – 93p2 + 6p – 9

=

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7. 2v – 32v 2 + v – 6

=

8. z + 2z2 + 4z + 4

=

9. g – 55 – g =

10. 2xy + 3xz– x =

11. 8k – hh 2 – 64k2 =

12. 2q2 + 17qr + 21r 2

3q2 + 26qr + 35r 2 =

13. 5u – u 2

3u 3 – 9u 2 – 30u =

14. –s2 + 7st – 12t2

s2 – 5st + 6t2 =

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1.3 SUMMARY ACTIVITY

1. What is an algebraic fraction?

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

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

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

2. Explain the two steps in the algorithm for reducing an algebraic fraction to

lowest terms.

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

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

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

3. A mistake was made in the simplification of each algebraic fraction shown

below. In each case, explain why the simplification is wrong.

a) 2x – 3y2y – 3x = 2x – 3y

2y – 3x = 1

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

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

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

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b) n + 1n + 2 = n + 1

n + 2 = 12

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

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

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

c) a – 3b – 3

= a + 3b + 3

= ab

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

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

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

d) 5t – 7u7u – 5t = 5t – 7u

7u – 5t = 1

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

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

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

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1.4 THE MATH WHIZ PAGE

Take a Bite Out of These Problems!

Here are five brain teasers.

Reduce the following algebraic fractions to lowest terms.

1. (3x 2 + 7xy + 2y 2)(2x 2 + 15xy + 28y 2)(2x 2 + 11xy + 14y 2)(3x 2 + 13xy + 4y 2)

=

mth-4110-1 the f operations to be met. i see!? start lastly, the finish line indicates ......

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