The Four Operations Algebraic Fractions

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

MTH-4110-1

he Four Operations on AlgebraicT 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

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 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. Simplifying Algebraic Fractions ...... 1.1 2. Product and Quotient of Algebraic Fractions ...... 2.1 3. Multiplying and Dividing Algebraic Fractions ...... 3.1 4. Adding and Subtracting Algebraic Fractions ...... 4.1 5. 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

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 © SOFAD MTH-4110-1 The Four Operations on Algebraic Fractions

THE PROGRAM FLOWCHART CEGEP

MTH-5112-1 Logic

MTH-5111-2 Complement and Synthesis II MTH-536 MTH-5110-1 Introduction to Vectors

MTH-5109-1 Geometry IV

MTH-5108-1 Trigonometric Functions and Equations MTH-526 Exponential and Logarithmic Functions MTH-5107-1 and Equations MTH-5104-1 Optimization II MTH-5106-1 Real Functions and Equations MTH-514 MTH-5103-1 Probability II MTH-5105-1 Conics

MTH-5102-1 Statistics III MTH-5101-1 Optimization I

MTH-4111-2 Complement and Synthesis I MTH-436 Trades The Four Operations on MTH-4110-1 You ar e here DVS Algebraic Fractions

MTH-4109-1 Sets, Relations and Functions MTH-4108-1 Quadratic Functions MTH-426 MTH-4107-1 Straight Lines II MTH-4106-1 Factoring and Algebraic Functions MTH-4105-1 Exponents and Radicals

MTH-4104-2 Statistics II

MTH-4103-1 Trigonometry I MTH-416 MTH-4102-1 Geometry III MTH-4101-2 Equations and Inequalities II

MTH-3003-2 Straight Lines I

MTH-314 MTH-3002-2 Geometry II MTH-3001-2 The Four Operations on Polynomials

MAT-2008-2 Statistics and Probabilities I MTH-216 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-116 MTH-1005-2 The Four Operations on Integers 25 hours = 1 credit

50 hours = 2 credits

HOW TO USE THIS GUIDE

Hi! My name is Monica and I have been You’ll see that with this method, math is I’m Andy. asked to tell you about this math module. a real breeze! What’s your name?

My results on the test Whether you are ... you have probably taken a placement test which tells you indicate that I should begin registered at an with this module. adult education exactly which module you center or at should start with. Formation à distance, ...

Now, the module you have in your ... the entry activity, which By carefully correcting this test using the hand is divided into three contains the test on the corresponding answer key, and record- sections. The first section is... prerequisites. ing your results on the analysis sheet ...

0.6 © SOFAD MTH-4110-1 The Four Operations on Algebraic Fractions

... you can tell if you’re well enough And if I’m not, if I need a little In that case, before you start the prepared to do all the activities in the activities in the module, the results review before moving on, what analysis chart refers you to a review module. happens then? activity near the end of the module.

I see!

In this way, I can be sure I have all the prerequisites START for starting. The starting line shows where the Exactly! The second section contains the learning activities. It’s learning activities the main part of the module. begin.

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.

The boldface question mark indicates practice exercices which allow you to try out what ? you have just learned. Look closely at the box to the right. It explains the symbols used to identify the various activities. 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.

There are also many fun things in this module. For example, A “Did you know that...”? Must I memorize what the sage says? when you see the drawing of a sage, it introduces a “Did you know that...” Yes, for example, short tidbits No, it’s not part of the learn- on the history of mathematics ing activity. It’s just there to and fun puzzles. They are in- give you a breather. teresting and relieve tension at the same time.

It’s the same for the “math whiz” pages, which are designed espe- They are so stimulating that And the whole module has cially for those who love math. even if you don’t have to do been arranged to make them, you’ll still want to. learning easier.

For example. words in bold- ... statements in boxes are important The third section contains the final re- face italics appear in the points to remember, like definitions, for- view, which interrelates the different glossary at the end of the mulas and rules. I’m telling you, the for- parts of the module. module... mat makes everything much easier. Great!

There is also a self-evaluation Thanks, Monica, you’ve been a big Later ... test and answer key. They tell help. you if you’re ready for the final I’m glad! Now, This is great! I never thought that I would evaluation. I’ve got to run. like mathematics as much as this! See you!

0.8 © SOFAD MTH-4110-1 The Four Operations on Algebraic Fractions

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 2 2 2 p q x +6x +8, 4ab , 1 and 2x +4 9 9m2 –4n 2 2p2q2 –5pq 2 –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 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.

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.

0.10 © SOFAD MTH-4110-1 The Four Operations on Algebraic Fractions

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* % (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.

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.

0.12 © SOFAD MTH-4110-1 The Four Operations on Algebraic Fractions

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.

1. Reduce each fraction to lowest terms.

a) 60 = b) 26 = 80 ...... 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) 9 × 2 = 10 45 ......

b) 3 × 14 × 5 = 7 15 8 ......

c) 3 ÷ 7 = 4 8 ......

d) 2 ÷ 2 = 9 3 ......

0.14 © SOFAD MTH-4110-1 The Four Operations on Algebraic Fractions

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) = ......

2 f) 1 x2 – 1 = 3 2 ......

3 2 g) 25a b +(3a2 +2b)2b = 5ab ......

h) 1 xy 1 x2 y – 1 xy + 2 x2 y ÷ 2 xy = 2 3 2 3 3

......

0.16 © SOFAD MTH-4110-1 The Four Operations on Algebraic Fractions

ANSWER KEY FOR THE DIAGNOSTIC TEST ON THE PREREQUISITES

× × 1. a) 60 = 3 20 = 3 b) 26 = 2 13 = 2 80 4 × 20 4 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 × 3. a) 9 × 2 = 1 1 = 1 10 45 5 × 5 25 5 5

1 1 2 1 × × b) 3 × 14 × 5 = 1 1 1 = 1 7 15 8 1 × 1 × 4 4 1 3 4 1

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

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

4. a) 3 + 7 = 12 + 7 = 19 b) 2 + 3 = 14 + 15 = 29 8 32 32 32 32 5 7 35 35 35

c) 5 – 7 = 25 – 14 = 11 d) 1 – 1 = 10 – 13 = –3 6 15 30 30 30 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

2 2 2 f) 1 x2 – 1 = 1 x2 –2 1 x2 × 1 + –1 = 1 x4 – 1 x2 + 1 3 2 3 3 2 2 9 3 4

3 2 g) 25a b +(3a2 +2b)2b =5a2b +6a2b +4b2 =11a2b +4b2 5ab

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

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

0.18 © SOFAD MTH-4110-1 The Four Operations on Algebraic Fractions

ANALYSIS OF THE DIAGNOSTIC TEST RESULTS

Answer Review Before Going Question Correct Incorrect Section Page to Unit(s) 1. a) 7.1 7.4 1 to 5 b) 7.1 4.4 1 to 5 2. a) 7.2 7.7 1 to 5 b) 7.2 7.7 1 to 5 c) 7.2 7.7 1 to 5 d) 7.2 7.7 1 to 5 e) 7.2 7.7 1 to 5 f) 7.2 7.7 1 to 5 g) 7.2 7.7 1 to 5 h) 7.2 7.7 1 to 5 i) 7.2 7.7 1 to 5 j) 7.2 7.7 1 to 5 3. a) 7.3 7.23 2, 3 and 5 b) 7.3 7.23 2, 3 and 5 c) 7.3 7.23 2, 3 and 5 d) 7.3 7.23 2, 3 and 5 4. a) 7.4 7.28 4 and 5 b) 7.4 7.28 4 and 5 c) 7.4 7.28 4 and 5 d) 7.4 7.28 4 and 5 5. a) 7.5 7.37 3 to 5 b) 7.5 7.37 3 to 5 c) 7.5 7.37 3 to 5 d) 7.5 7.37 3 to 5 e) 7.5 7.37 3 to 5 f) 7.5 7.37 3 to 5 g) 7.5 7.37 3 to 5 h) 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).

MTH-4110-1 The Four Operations on Algebraic Fractions

INFORMATION FOR DISTANCE EDUCATION STUDENTS

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.

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.

0.22 © SOFAD MTH-4110-1 The Four Operations on Algebraic Fractions

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.

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.

0.24 © SOFAD MTH-4110-1 The Four Operations on Algebraic Fractions

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

You probably obtained the following answers: 1 , 1 , 1 and 2 . 2 14 11 19

4 = 1 13 To arrive at these answers, you had to 8 2 = 1 143 11 5 1 = 18 = 1 find the greatest common factor of the 70 14 171 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?

2 3 4x , 42ab c , 2m +8 8x 2 3a2b 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.

1.2 © SOFAD MTH-4110-1 The Four Operations on Algebraic Fractions

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: 2 3 4x , 42ab c , 2m +8 8x 2 3a2b m2 +6m +8

× 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.

2 3 3 × 3 b) 42ab c = 14bc 3ab = 14bc 3a2b a × 3ab a

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

2 × (m +4) c) 2m +8 = = 2 m2 +6m +8 (m +2)× (m +4) m +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).

1.4 © SOFAD MTH-4110-1 The Four Operations on Algebraic Fractions

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

2 p –4 a) Simplify the algebraic fraction . 2p2 +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)

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

2 p –4 This is the simplest expression of the fraction . 2p2 +7p +6

b) Simplify the algebraic fraction 2x –14 . 3x –21+bx –7b

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

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.

6g 1. = 9g

2 5 2. 15s tu = 27stu 2

2 2 3. a – a b = c 3 – c 3b

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

2 5. v – v –12 = v –4

6. 6m –12 = 8m –16

2 7. ab + bc + a + ac = b2 – a2

2 8. r –2r –15 = 4r 2 +13r +3

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 a + b 2 2 fraction 2 , the term b cannot be cancelled out, as b is not a factor of 2 b 2 the polynomial a2 + b2. In other words, a + b ≠ a2 . 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.

Law of signs for multiplication and division: + × +=+ +÷+=+ – × –=+ and–÷–=+ + × –=– +÷–=– – × +=– –÷+=–

Example 3

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

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

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

y +1 –( y +1) 1– y The answer is – y or y or y .

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 +8rs +12s cannot be simplified because, after r 2 +7rs +12s 2 (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.

Exercise 1.2

Reduce the following algebraic fractions to lowest terms.

2 2 2

1. h – h k = h 3 – h 3k2

2. ab – bc = – b

4– j 3. = j 2 –16

4x 2 –8xy –12y 2 4. = 3y – x

5. 21 m –3n = n 2 –49m2

6. 2d –6 = 2d 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.

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:

In both cases, the executioner cannot execute you... Whew! you... execute cannot executioner the cases, both In

is the truth, and the punishment is again wrong. again is punishment the and truth, the is

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

is a lie, and the punishment is wrong. is punishment the and lie, a is

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

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

? 1.2 PRACTICE EXERCISES

Reduce the algebraic fractions below to lowest terms.

6 x 2 –8xy 1. = 9xy –12y 2

2 3 2. 15a b = –3a3b

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

2 2 4. c –4d = 2d + c

(2 j +6)2 5. = 4 j 2 –36

3p –9 6. = 3p2 +6p –9

7. 2v –3 = 2v 2 + v –6

8. z +2 = z 2 +4z +4

g –5 9. = 5– g

2xy +3xz 10. – x =

11. 8 k – h = h 2 –64k2

2q 2 +17qr +21r 2 12. = 3q2 +26qr +35r 2

2 13. 5u – u = 3u 3 –9u 2 –30u

2 2 14. –s +7st –12t = s 2 –5st +6t 2

2 15. 4t –36t +80 = (4t – t 2)(5 – t)

2y 2 –4yz +2z 2 16. = 10x 2 y –10yz 2

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.

2x –3y 2x –3y a) = = 1 2y –3x 2y –3x

......

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

......

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.

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

24 a3c 2d 2(2d 2 – d –3) 2. (18a2d 2 –12a2d 3)(1 – d 2)

4–(m + n) 2 3. 8(2 + m + n)(2 – m – n)

(r – s) 2 – t 2 4. = r 2 –(s – t) 2

5. h + hk = h 2 – k2 – h – k