MTH-4110-1 C1

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MTH-4110-1

he Four Operations on

Algebraic Fractions

MTH-4110-1 THE FOUR OPERATIONS ON ALGEBRAIC FRACTIONS

Author: Suzie Asselin Content revision: Daniel Gélineau Jean-Paul Groleau Mireille Moisan-Sanscartier Nicole 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 langue anglaise 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 — 2004 Bibliothèque et Archives nationales du Québec Bibliothèque et Archives Canada ISBN 2-89493-288-9

Answer Key

MTH-4110-1

The Four Operations on Algebraic Fractions

TABLE OF CONTENTS

Introduction to the Program Flowchart ................................................... 0.4 The Program Flowchart ............................................................................ 0.5 How to Use This Guide ............................................................................. 0.6 General Introduction ................................................................................. 0.9 Intermediate and Terminal Objectives of the Module ............................ 0.11 Diagnostic Test on the Prerequisites ....................................................... 0.13 Answer Key for the Diagnostic Test on the Prerequisites ...................... 0.17 Analysis of Diagnostic Test Results ......................................................... 0.19 Information for Distance Education Students ......................................... 0.21 UNITS 1. 2. 3. 4. 5.

Simplifying Algebraic Fractions ............................................................... 1.1 Product and Quotient of Algebraic Fractions .......................................... 2.1 Multiplying and Dividing Algebraic Fractions ........................................ 3.1 Adding and Subtracting Algebraic Fractions .......................................... 4.1 Order of Operations Involving Algebraic Fractions ................................ 5.1 Final Summary.......................................................................................... 6.1 Answer Key for the Final Summary ........................................................ 6.5 Terminal Objective .................................................................................... 6.6 Self-Evaluation Test.................................................................................. 6.7 Answer Key for the Self-Evaluation Test ................................................ 6.13 Analysis of the Self-Evaluation Test Results .......................................... 6.17 Final Evaluation........................................................................................ 6.18 Answer Key for the Exercises ................................................................... 6.19 Glossary ..................................................................................................... 6.41 List of Symbols .......................................................................................... 6.45 Bibliography .............................................................................................. 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! 0.4

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THE PROGRAM FLOWCHART

CEGEP MTH-5112-1 MTH-5111-2

MTH-536

MTH-5104-1 MTH-5103-1

Introduction to Vectors

MTH-5109-1

Geometry IV

MTH-5108-1

Trigonometric Functions and Equations

MTH-5107-1

Exponential and Logarithmic Functions and Equations

Optimization II

MTH-5106-1

Real Functions and Equations

Probability II

MTH-5105-1

Conics

MTH-5102-1

Statistics III

MTH-5101-1

MTH-436

MTH-426

MTH-4110-1

MTH-216

MTH-116

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The Four Operations on Algebraic Fractions

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 MTH-4102-1 MTH-4101-2

Complement and Synthesis I

MTH-4109-1

MTH-4104-2

MTH-314

Optimization I

MTH-4111-2

Trades DVS

MTH-416

Complement and Synthesis II

MTH-5110-1

MTH-526

MTH-514

Logic

You ar e h er e

Statistics II Trigonometry I Geometry III Equations and Inequalities II

MTH-3003-2

Straight Lines I

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

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25 hours

= 1 credit

50 hours

= 2 credits

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HOW TO USE THIS GUIDE

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

Whether you are registered at an adult education center or at Formation à distance, ...

Now, the module you have in your hand is divided into three sections. The first section is...

I’m Andy.

... you have probably taken a placement test which tells you exactly which module you should start with.

... the entry activity, which contains the test on the prerequisites.

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

My results on the test indicate that I should begin with this module.

By carefully correcting this test using the corresponding answer key, and recording your results on the analysis sheet ...

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... you can tell if you’re well enough prepared to do all the activities in the module.

And if I’m not, if I need a little review before moving on, what happens then?

In that case, before you start the activities in the module, the results analysis chart refers you to a review activity near the end of the module.

I see!

In this way, I can be sure I have all the prerequisites for starting.

START

The starting line shows where the learning activities begin.

Exactly! The second section contains the learning activities. It’s the main part of the module.

?

The little white question mark indicates the questions for which answers are given in the text. The target precedes the objective to be met. The memo pad signals a brief reminder of concepts which you have already studied.

? Look closely at the box to the right. It explains the symbols used to identify the various activities.

The boldface question mark indicates practice exercices which allow you to try out what you have just learned.

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

?

The sheaf of wheat indicates a review designed to reinforce what you have just learned. A row of sheaves near the end of the module indicates the final review, which helps you to interrelate all the learning activities in the module. FINISH

Lastly, the finish line indicates that it is time to go on to the self-evaluation test to verify how well you have understood the learning activities.

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There are also many fun things in this module. For example, when you see the drawing of a sage, it introduces a “Did you know that...”

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

For example. words in boldface italics appear in the glossary at the end of the module...

A “Did you know that...”? Yes, for example, short tidbits on the history of mathematics and fun puzzles. They are interesting and relieve tension at the same time.

Must I memorize what the sage says? No, it’s not part of the learning activity. It’s just there to give you a breather.

They are so stimulating that even if you don’t have to do them, you’ll still want to.

And the whole module has been arranged to make learning easier.

... statements in boxes are important points to remember, like definitions, formulas and rules. I’m telling you, the format makes everything much easier.

The third section contains the final review, which interrelates the different parts of the module.

Great!

There is also a self-evaluation test and answer key. They tell you if you’re ready for the final evaluation.

Thanks, Monica, you’ve been a big help. I’m glad! Now, I’ve got to run. See you!

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Later ... This is great! I never thought that I would like 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 2 p 2q 2 x 2 + 6x + 8 , 4ab , 1 and 9 2x + 4 9m 2 – 4n 2 2 p 2q 2 – 5 pq 2 – 3q 2

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 division 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. © SOFAD

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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 OF THE 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*

1 to 5

24

% (evaluation) 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 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, answers must be identical to those in the key. In addition, the various steps in your answer should be equivalent to those shown in the solution. 7. Transcribe your results onto the chart which follows the answer key. It gives an analysis of the diagnostic test results. 8. Do only the review activities which are suggested for each of your incorrect answers. 9. If all your answers are correct, you may begin working on this module.

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1. Reduce each fraction to lowest terms. a) 60 = 80 ..........................................

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. 9 × 2 = 10 45 .................................................................................................... b) 3 × 14 × 5 = 7 15 8 ............................................................................................... c) 3 ÷ 7 = 4 8 ........................................................................................................ d) 2 ÷ 2 = 9 3 ........................................................................................................

a)

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4. Perform the following additions and subtractions. Your results must be reduced to lowest terms. a) 3 + 7 = 8 32 ...................................................................................................... b) 2 + 3 = 5 7 ........................................................................................................ c) 5 – 7 = 6 15 ...................................................................................................... d) 1 – 1 = 13 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)

1 x2 – 1 2 3

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3 2 g) 25a b + (3a 2 + 2b)2b = 5ab ..............................................................................

h)

1 xy 1 x 2 y – 1 xy + 2 x 2 y 3 2 3 2

÷ 2 xy = 3

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

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ANSWER KEY FOR THE DIAGNOSTIC TEST ON THE PREREQUISITES

1. a) 60 = 3 × 20 = 3 80 4 × 20 4

b) 26 = 2 × 13 = 2 65 5 × 13 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)

1

1

5

5

3. a) 9 × 2 = 1 × 1 = 1 10 45 5 × 5 25 1 1

2

1

1

3 1

4

b) 3 × 14 × 5 = 1 × 1 × 1 = 1 7 15 8 1 × 1 × 4 4

2

c) 3 ÷ 7 = 3 × 8 = 3 × 2 = 6 4 8 4 7 1×7 7 1

1

1

3

1

d) 2 ÷ 2 = 2 × 3 = 1 × 1 = 1 9 3 9 2 3×1 3

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4. a) 3 + 7 = 12 + 7 = 19 8 32 32 32 32

b) 2 + 3 = 14 + 15 = 29 5 7 35 35 35

c) 5 – 7 = 25 – 14 = 11 6 15 30 30 30

d)

1 – 1 = 10 – 13 = – 3 13 10 130 130 130

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)

1 x2 – 1 2 3

2

= 1 x2 3

2

– 2 1 x 2 × 1 + –1 3 2 2

2

= 1 x4 – 1 x2 + 1 9 3 4

3 2 g) 25a b + (3a 2 + 2b)2b = 5a 2b + 6a 2b + 4b 2 = 11a 2b + 4b 2 5ab

h)

1 xy 1 x 2 y – 1 xy + 2 x 2 y 3 2 3 2

2xy ÷ 2 xy = 1 x 3 y 2 – 1 x 2 y 2 + 1 x 3 y 2 ÷ = 3 6 4 3 3

x 3y 2 x 2 y 2 3x 2 y 3xy – × 3 = – or 3 x 2 y – 3 xy 2 4 2xy 4 8 8 4

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ANALYSIS OF THE DIAGNOSTIC TEST RESULTS Question 1. a) b) 2. a) b) c) d) e) f) g) h) i) j) 3. a) b) c) d) 4. a) b) c) d) 5. a) b) c) d) e) f) g) h)

Answer Correct

Incorrect

Section

Review Page

Before Going to Unit(s)

7.1 7.1 7.2 7.2 7.2 7.2 7.2 7.2 7.2 7.2 7.2 7.2 7.3 7.3 7.3 7.3 7.4 7.4 7.4 7.4 7.5 7.5 7.5 7.5 7.5 7.5 7.5 7.5

7.4 4.4 7.7 7.7 7.7 7.7 7.7 7.7 7.7 7.7 7.7 7.7 7.23 7.23 7.23 7.23 7.28 7.28 7.28 7.28 7.37 7.37 7.37 7.37 7.37 7.37 7.37 7.37

1 to 5 1 to 5 1 to 5 1 to 5 1 to 5 1 to 5 1 to 5 1 to 5 1 to 5 1 to 5 1 to 5 1 to 5 2, 3 and 5 2, 3 and 5 2, 3 and 5 2, 3 and 5 4 and 5 4 and 5 4 and 5 4 and 5 3 to 5 3 to 5 3 to 5 3 to 5 3 to 5 3 to 5 3 to 5 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). © SOFAD

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INFORMATION FOR EDUCATION STUDENTS

DISTANCE

You now have the learning material for MTH-4110-1 together with the homework 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 assignments 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 pertinent 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. © SOFAD

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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 assignment, 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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START

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 4 is ............. . 8

?

If we simplify the fraction 5 , we get ............... . 70

?

By reducing 13 to lowest terms, we obtain ............... . 143

?

After simplifcation, the fraction 18 becomes ............... . 171

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You probably obtained the following answers: 1 , 1 , 1 and 2 . 19 2 14 11 To arrive at these answers, you had to find the greatest common factor of the

4= 8 5 = 1 70 14

1 2

13 = 1 143 11 18 = 1 171 19

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 a , the term a is called the b numerator and the term b is called the denominator. Do you know how to simplify algebraic fractions like these ones? 2m + 8 4x , 42ab 2c 3 , 3a 2b m 2 + 6m + 8 8x 2

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: 2m + 8 4x , 42ab 2c 3 , 2 2 2 3a b m + 6m + 8 8x

a)

4x = 1 × 4x = 1 8x 2 2x × 4x 2x

• The algebraic expression 4x is the greatest common factor of both the numerator and the denominator.

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2 3 3 3 b) 42ab2 c = 14bc × 3ab = 14bc a a × 3ab 3a b

• 3ab is the greatest common factor of the numerator and the denominator.

c)

2 × (m + 4) 2m + 8 = = 2 m + 6m + 8 (m + 2) × (m + 4) m + 2 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 themselves 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 – 4 . 2 p2 + 7 p + 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) (2 p + 3)( p + 2)

2. The common factor of the numerator and the denominator is (p + 2). Remove this factor: ( p – 2)( p + 2) p– 2 = (2 p + 3)( p + 2) 2 p + 3

This is the simplest expression of the fraction b) Simplify the algebraic fraction

p2 – 4 . 2 p2 + 7 p + 6

2x – 14 . 3x – 21 + bx – 7b

2(x – 7) 2x – 14 = = 2 3x – 21 + bx – 7b (x – 7)(3 + b) 3 + 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 denominator. 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.

6g = 9g

5 2 2. 15s tu2 = 27stu

2 2 3. a 3 – a3 b = c –c b

4.

2x + 6 = x + 5x + 6 2

2 5. v – v – 12 = v–4

6. 6m – 12 = 8m – 16

2 7. ab + bc2 + a2 + ac = b –a

8.

r 2 – 2r – 15 = 4r 2 + 13r + 3

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N.B. If you had difficulty solving the preceding problems, reread the explanations 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 2 2 fraction a +2 b , the term b2 cannot be cancelled out, as b2 is not a factor of 2 b 2 the polynomial a2 + b2. In other words, a + b ≠ a 2 . b2 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

y2 – 1 . y – y2

1. Factor the numerator and the denominator. ( y + 1)( y – 1) ( y + 1)( y – 1) ( y + 1)( y – 1) = = – y(–1 + y) – y( y – 1) y(1 – y)

2. Simplify by removing the common factor. ( y + 1)( y – 1) – y( y – 1)

The answer is

– ( y + 1) y+1 1– y or y . y – y or

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 2 2 The algebraic fraction r 2 + 8rs + 12s 2 cannot be simplified because, after r + 7rs + 12s (r + 6s)(r + 2s) factoring, we obtain the fraction , which does not contain any (r + 4s)(r + 3s)

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. 2 2 2 1. h 3 – h 3k 2 = h –h k

2. ab – bc = –b

3.

4– j = 2 j – 16

4.

4x 2 – 8xy – 12 y 2 = 3y – x

5.

21m – 3n = n 2 – 49m 2

6.

2d – 6 = 2d – 7d + 6 2

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:

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!

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

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PRACTICE EXERCISES

Reduce the algebraic fractions below to lowest terms.

1.

6x 2 – 8xy = 9xy – 12 y 2

2 3 2. 15a 3b = – 3a b

3. 2m + 2n2 = (m + n)

2 2 4. c – 4d = 2d + c

5.

(2 j + 6) 2 = 2 4 j – 36

6.

3p – 9 = 3p + 6p – 9 2

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2v – 3 = 2v 2 + v – 6

8.

z+2 = z + 4z + 4

9.

g–5 = 5– g

The Four Operations on Algebraic Fractions

2

10.

2xy + 3xz = –x

11.

8k – h = h 2 – 64k 2

12.

2q 2 + 17qr + 21r 2 = 3q 2 + 26qr + 35r 2

13.

5u – u 2 = 3u 3 – 9u 2 – 30u

2 2 14. – s2 + 7st – 12t = s – 5st + 6t 2

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2 + 80 = 15. 4t – 36t 2 (4t – t )(5 – t)

16.

2y 2 – 4 yz + 2z 2 = 10x 2 y – 10 yz 2

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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 – 3y 2x – 3y =1 = 2y – 3x 2y – 3x

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

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

....................................................................................................................... ....................................................................................................................... ....................................................................................................................... c) a – 3 = a + 3 = a b–3 b+3 b

....................................................................................................................... ....................................................................................................................... ....................................................................................................................... d) 5t – 7u = 5t – 7u = 1 7u – 5t 7u – 5t

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

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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 + 28 y 2) = (2x 2 + 11xy + 14 y 2)(3x 2 + 13xy + 4 y 2)

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2.

24a 3c 2d 2(2d 2 – d – 3) (18a 2d 2 – 12a 2d 3)(1 – d 2)

3.

4 – (m + n) 2 8(2 + m + n)(2 – m – n)

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2

4.

(r – s) – t 2 2 = r 2 – (s – t)

5.

h 2 + hk = h 2 – k2 – h – k

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