Cambridge 3 unit Mathematics

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YEAR

11

CAMBRIDGE

Mathematics

BILL PENDER

DAVID SADLER

JULIA SHEA

DEREK WARD

Enhanced

3 Unit

Extension 1

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cambridge university press

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Bill Pender, David Sadler, Julia Shea, Derek Ward 2012 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press.

First published 1999 Reprinted 2001, 2002, 2008

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iv Contents

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Chapter Four—Trigonometry . . . 107

4A Trigonometry with Right Triangles . . . 107

4B Theoretical Exercises on Right Triangles . . . 114

4C Trigonometric Functions of a General Angle . . . 117

4D The Quadrant, the Related Angle and the Sign . . . 121

4E Given One Trigonometric Function, Find Another . . . 127

4F Trigonometric Identities and Elimination . . . 129

4G Trigonometric Equations . . . 133

4H The Sine Rule and the Area Formula . . . 139

4I The Cosine Rule . . . 146

4J Problems Involving General Triangles . . . 150

Chapter Five—Coordinate Geometry . . . 156

5A Points and Intervals . . . 156

5B Gradients of Intervals and Lines . . . 162

5C Equations of Lines . . . 167

5D Further Equations of Lines . . . 170

5E Perpendicular Distance . . . 176

5F Lines Through the Intersection of Two Given Lines . . . 180

5G Coordinate Methods in Geometry . . . 184

Chapter Six—Sequences and Series . . . 188

6A Indices . . . 188

6B Logarithms . . . 192

6C Sequences and How to Specify Them . . . 196

6D Arithmetic Sequences . . . 200

6E Geometric Sequences . . . 203

6F Arithmetic and Geometric Means . . . 207

6G Sigma Notation . . . 211

6H Partial Sums of a Sequence . . . 213

6I Summing an Arithmetic Series . . . 215

6J Summing a Geometric Series . . . 219

6K The Limiting Sum of a Geometric Series . . . 223

6L Recurring Decimals and Geometric Series . . . 227

6M Factoring Sums and Diﬀerences of Powers . . . 229

6N Proof by Mathematical Induction . . . 231

Chapter Seven—The Derivative . . . 237

7A The Derivative — Geometric Deﬁnition . . . 237

7B The Derivative as a Limit . . . 241

7C A Rule for Diﬀerentiating Powers of x . . . 245

7D The Notation dy dx for the Derivative . . . 250

7E The Chain Rule . . . 254

7F The Product Rule . . . 260

7G The Quotient Rule . . . 262

7H Rates of Change . . . 265

7I Limits and Continuity . . . 268

7J Diﬀerentiability . . . 273

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Contents v

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Chapter Eight—The Quadratic Function . . . 280

8A Factorisation and the Graph . . . 280

8B Completing the Square and the Graph . . . 285

8C The Quadratic Formulae and the Graph . . . 289

8D Equations Reducible to Quadratics . . . 292

8E Problems on Maximisation and Minimisation . . . 294

8F The Theory of the Discriminant . . . 299

8G Deﬁnite and Indeﬁnite Quadratics . . . 304

8H Sum and Product of Roots . . . 307

8I Quadratic Identities . . . 311

Chapter Nine—The Geometry of the Parabola . . . 316

9A A Locus and its Equation . . . 316

9B The Geometric Deﬁnition of the Parabola . . . 320

9C Translations of the Parabola . . . 325

9D Parametric Equations of Curves . . . 327

9E Chords of a Parabola . . . 330

9F Tangents and Normals: Parametric Approach . . . 333

9G Tangents and Normals: Cartesian Approach . . . 338

9H The Chord of Contact . . . 341

9I Geometrical Theorems about the Parabola . . . 345

9J Locus Problems . . . 350

Chapter Ten—The Geometry of the Derivative . . . 357

10A Increasing, Decreasing and Stationary at a Point . . . 357

10B Stationary Points and Turning Points . . . 362

10C Critical Values . . . 367

10D Second and Higher Derivatives . . . 371

10E Concavity and Points of Inﬂexion . . . 373

10F Curve Sketching using Calculus . . . 378

10G Global Maximum and Minimum . . . 380

10H Applications of Maximisation and Minimisation . . . 383

10I Maximisation and Minimisation in Geometry . . . 388

10J Primitive Functions . . . 391

Chapter Eleven—Integration . . . 397

11A Finding Areas by a Limiting Process . . . 397

11B The Fundamental Theorem of Calculus . . . 402

11C The Deﬁnite Integral and its Properties . . . 407

11D The Indeﬁnite Integral . . . 412

11E Finding Area by Integration . . . 415

11F Area of a Compound Region . . . 419

11G Volumes of Solids of Revolution . . . 423

11H The Reverse Chain Rule . . . 429

11I The Trapezoidal Rule . . . 432

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vi Contents

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Chapter Twelve—The Logarithmic Function . . . 438

12A Review of Logarithmic and Exponential Functions . . . 438

12B The Logarithmic Function and its Derivative . . . 441

12C Applications of Diﬀerentiation . . . 450

12D Integration of the Reciprocal Function . . . 454

12E Applications of Integration . . . 459

Chapter Thirteen—The Exponential Function . . . 462

13A The Exponential Function and its Derivative . . . 462

13B Applications of Diﬀerentiation . . . 467

13C Integration of the Exponential Function . . . 472

13D Applications of Integration . . . 476

13E Natural Growth and Decay . . . 479

Chapter Fourteen—The Trigonometric Functions . . . 487

14A Radian Measure of Angle Size . . . 487

14B Mensuration of Arcs, Sectors and Segments . . . 491

14C Graphs of the Trigonometric Functions in Radians . . . 496

14D Trigonometric Functions of Compound Angles . . . 504

14E The Angle Between Two Lines . . . 509

14F The Behaviour of sin x Near the Origin . . . 513

14G The Derivatives of the Trigonometric Functions . . . 517

14H Applications of Diﬀerentiation . . . 523

14I Integration of the Trigonometric Functions . . . 528

14J Applications of Integration . . . 534

Answers to Exercises . . . 538

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Preface

This textbook has been written for students in Years 11 and 12 taking the course previously known as ‘3 Unit Mathematics’, but renamed in the new HSC as two courses, ‘Mathematics’ (previously called ‘2 Unit Mathematics’) and ‘Mathemat-ics, Extension 1’. The book develops the content at the level required for the 2 and 3 Unit HSC examinations. There are two volumes — the present volume is roughly intended for Year 11, and the second for Year 12. Schools will, however, diﬀer in their choices of order of topics and in their rates of progress.

Although these Syllabuses have not been rewritten for the new HSC, there has been a gradual shift of emphasis in recent examination papers.

• The interdependence of the course content has been emphasised. • Graphs have been used much more freely in argument.

• Structured problem solving has been expanded. • There has been more stress on explanation and proof.

This text addresses these new emphases, and the exercises contain a wide variety of diﬀerent types of questions.

There is an abundance of questions in each exercise — too many for any one student — carefully grouped in three graded sets, so that with proper selection the book can be used at all levels of ability. In particular, those who subse-quently drop to 2 Units of Mathematics, and those who in Year 12 take 4 Units of Mathematics, will both ﬁnd an appropriate level of challenge. We have written a separate book, also in two volumes, for the 2 Unit ‘Mathematics’ course alone. We would like to thank our colleagues at Sydney Grammar School and Newington College for their invaluable help in advising us and commenting on the successive drafts, and for their patience in the face of some diﬃculties in earlier drafts. We would also like to thank the Head Masters of Sydney Grammar School and Newington College for their encouragement of this project, and Peter Cribb and the team at Cambridge University Press, Melbourne, for their support and help in discussions. Finally, our thanks go to our families for encouraging us, despite the distractions it has caused to family life.

Preface to the enhanced version

To provide students with practice for the new objective response (multiple choice) questions to be included in HSC examinations, online self-marking quizzes have been provided for each chapter, on Cambridge GO (access details can be found in the following pages). In addition, an interactive textbook version is available through the same website.

Dr Bill Pender

Subject Master in Mathematics Sydney Grammar School College Street

Darlinghurst NSW 2010 David Sadler

Mathematics

Sydney Grammar School

Julia Shea Head of Mathematics Newington College 200 Stanmore Road Stanmore NSW 2048 Derek Ward Mathematics

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How to Use This Book

This book has been written so that it is suitable for the full range of 3 Unit students, whatever their abilities and ambitions. The book covers the 2 Unit and 3 Unit content without distinction, because 3 Unit students need to study the 2 Unit content in more depth than is possible in a 2 Unit text. Nevertheless, students who subsequently move to the 2 Unit course should ﬁnd plenty of work here at a level appropriate for them.

The Exercises: No-one should try to do all the questions! We have written long

exercises so that everyone will ﬁnd enough questions of a suitable standard — each student will need to select from them, and there should be plenty left for revision. The book provides a great variety of questions, and representatives of all types should be selected.

Each chapter is divided into a number of sections. Each of these sections has its own substantial exercise, subdivided into three groups of questions:

Foundation: These questions are intended to drill the new content of the sec-tion at a reasonably straightforward level. There is little point in proceeding without mastery of this group.

Development: This group is usually the longest. It contains more substantial questions, questions requiring proof or explanation, problems where the new content can be applied, and problems involving content from other sections and chapters to put the new ideas in a wider context. Later questions here can be very demanding, and Groups 1 and 2 should be suﬃcient to meet the demands of all but exceptionally diﬃcult problems in 3 Unit HSC papers. Extension: These questions are quite hard. Some are algebraically

challeng-ing, some establish a general result beyond the theory of the course, some make diﬃcult connections between topics or give an alternative approach, some deal with logical problems unsuitable for the text of a 3 Unit book. Students taking the 4 Unit course should attempt some of these.

The Theory and the Worked Exercises: The theory has been developed with as much

rigour as is appropriate at school, even for those taking the 4 Unit course. This leaves students and their teachers free to choose how thoroughly the theory is presented in a particular class. It can often be helpful to learn a method ﬁrst and then return to the details of the proof and explanation when the point of it all has become clear.

The main formulae, methods, deﬁnitions and results have been boxed and num-bered consecutively through each chapter. They provide a summary only, and represent an absolute minimum of what should be known. The worked examples

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How to Use This Book xi

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have been chosen to illustrate the new methods introduced in the section, and should be suﬃcient preparation for the questions of the following exercise.

The Order of the Topics: We have presented the topics in the order we have found most satisfactory in our own teaching. There are, however, many effective or-derings of the topics, and the book allows all the flexibility needed in the many different situations that apply in different schools (apart from the few questions that provide links between topics).

The time needed for the algebra in Chapter One will depend on students’ expe-riences in Years 9 and 10. The same applies to other topics in the early chapters — trigonometry, quadratic functions, coordinate geometry and particularly curve sketching. The Study Notes at the start of each chapter make further speciﬁc remarks about each topic.

We have left Euclidean geometry and polynomials until Year 12 for two reasons. First, we believe as much calculus as possible should be developed in Year 11, ideally including the logarithmic and exponential functions and the trigonometric functions. These are the fundamental ideas in the course, and it is best if Year 12 is used then to consolidate and extend them (and students subsequently taking the 4 Unit course particularly need this material early). Secondly, the Years 9 and 10 Advanced Course already develops much of the work on polynomials and Euclidean geometry in Options recommended for those proceeding to 3 Unit, so that revisiting them in Year 12 with the extensions and far greater sophistication required seems an ideal arrangement.

The Structure of the Course: Recent examination papers have included longer ques-tions combining ideas from different topics, thus making clear the strong inter-connections amongst the various topics. Calculus is the backbone of the course, and the two processes of differentiation and integration, inverses of each other, dominate most of the topics. We have introduced both processes using geomet-rical ideas, basing differentiation on tangents and integration on areas, but the subsequent discussions, applications and exercises give many other ways of un-derstanding them. For example, questions about rates are prominent from an early stage.

Besides linear functions, three groups of functions dominate the course:

The Quadratic Functions: These functions are known from earlier years. They are algebraic representations of the parabola, and arise naturally in situations where areas are being considered or where a constant acceleration is being applied. They can be studied without calculus, but calculus provides an alternative and sometimes quicker approach.

The Exponential and Logarithmic Functions: Calculus is essential for the study of these functions. We have chosen to introduce the logarithmic function ﬁrst, using deﬁnite integrals of the reciprocal function y = 1/x. This approach is more satisfying because it makes clear the relationship between these functions and the rectangular hyperbola y = 1/x, and because it gives a clear picture of the new number e. It is also more rigorous. Later, however, one can never overemphasise the fundamental property that the exponential function with base e is its own derivative — this is the reason why these func-tions are essential for the study of natural growth and decay, and therefore occur in almost every application of mathematics.

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xii How to Use This Book

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Arithmetic and geometric sequences arise naturally throughout the course. They are the values, respectively, of linear and exponential functions at in-tegers, and these interrelationships need to be developed, particularly in the context of applications to ﬁnance.

The Trigonometric Functions: Again, calculus is essential for the study of these functions, whose deﬁnition, like the associated deﬁnition of π, is based on the circle. The graphs of the sine and cosine functions are waves, and they are essential for the study of all periodic phenomena — hence the detailed study of simple harmonic motion in Year 12.

Thus the three basic functions of the course — x2, exand sin x — and the related numbers e and π are developed from the three most basic degree 2 curves — the parabola, the rectangular hyperbola and the circle. In this way, everything in the course, whether in calculus, geometry, trigonometry, coordinate geometry or algebra, is easily related to everything else.

The geometry of the circle is mostly studied using Euclidean methods, and the highly structured arguments used here contrast with the algebraic arguments used in the coordinate geometry approach to the parabola. In the 4 Unit course, the geometry of the rectangular hyperbola is given special consideration in the context of a coordinate geometry treatment of general conics.

Polynomials are a generalisation of quadratics, and move the course a little be-yond the degree 2 phenomena described above. The particular case of the bi-nomial theorem then becomes the bridge from elementary probability using tree diagrams to the binomial distribution with all its practical applications. Unfor-tunately the power series that link polynomials with the exponential and trigono-metric functions are too sophisticated for a school course. Projective geometry and calculus with complex numbers are even further removed, so it is not really possible to explain that exponential and trigonometric functions are the same thing, although there are many clues.

Algebra, Graphs and Language: One of the chief purposes of the course, stressed in recent examinations, is to encourage arguments that relate a curve to its equation. Being able to predict the behaviour of a curve given only its equation is a constant concern of the exercises. Conversely, the behaviour of a graph can often be used to solve an algebraic problem. We have drawn as many sketches in the book as space allowed, but as a matter of routine, students should draw diagrams for almost every problem they attempt. It is because sketches can so easily be drawn that this type of mathematics is so satisfactory for study at school.

This course is intended to develop simultaneously algebraic agility, geometric intuition, and rigorous language and logic. Ideally then, any solution should

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How to Use This Book xiii

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display elegant and error-free algebra, diagrams to display the situation, and clarity of language and logic in argument.

Theory and Applications: Elegance of argument and perfection of structure are fun-damental in mathematics. We have kept to these values as far as is reasonable in the development of the theory and in the exercises. The application of mathe-matics to the world around us is an equally fundamental, and we have given many examples of the usefulness of everything in the course. Calculus is particularly suitable for presenting this double view of mathematics.

We would therefore urge the reader sometimes to pay attention to the details of argument in proofs and to the abstract structures and their interrelationships, and at other times to become involved in the interpretation provided by the applications.

Limits, Continuity and the Real Numbers: This is a ﬁrst course in calculus, geometri-cally and intuitively developed. It is not a course in analysis, and any attempt to provide a rigorous treatment of limits, continuity or the real numbers would be quite inappropriate. We believe that the limits required in this course present little diﬃculty to intuitive understanding — really little more is needed than

lim

x→∞1/x = 0 and the occasional use of the sandwich principle in proofs. Char-acterising the tangent as the limit of the secant is a dramatic new idea, clearly marking the beginning of calculus, and quite accessible. Continuity and differ-entiability need only occasional attention, given the well-behaved functions that occur in the course. The real numbers are defined geometrically as points on the number line, and provided that intuitive ideas about lines are accepted, ev-erything needed about them can be justified from this definition. In particular, the intermediate value theorem, which states that a continuous function can only change sign at a zero, is taken to be obvious.

These unavoidable gaps concern only very subtle issues of ‘foundations’, and we are fortunate that everything else in the course can be developed rigorously so that students are given that characteristic mathematical experience of certainty and total understanding. This is the great contribution that mathematics brings to all our education.

Technology: There is much discussion, but little agreement yet, about what role tech-nology should play in the mathematics classroom and which calculators or soft-ware may be eﬀective. This is a time for experimentation and diversity. We have therefore given only a few speciﬁc recommendations about technology, but we encourage such investigation, and to this version we have added some optional technology resources that can be accessed via the Cambridge GO website. The graphs of functions are at the centre of the course, and the more experience and intuitive understanding students have, the better able they are to interpret the mathematics correctly. A warning here is appropriate — any machine drawing of a curve should be accompanied by a clear understanding of why such a curve arises from the particular equation or situation.

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About the Authors

Dr Bill Pender is Subject Master in Mathematics at Sydney Grammar School, where he has taught since 1975. He has an MSc and PhD in Pure Mathematics from Sydney University and a BA (Hons) in Early English from Macquarie Uni-versity. In 1973–4, he studied at Bonn University in Germany and he has lectured and tutored at Sydney University and at the University of NSW, where he was a Visiting Fellow in 1989. He was a member of the NSW Syllabus Committee in Mathematics for two years and subsequently of the Review Committee for the Years 9–10 Advanced Syllabus. He is a regular presenter of inservice courses for AIS and MANSW, and plays piano and harpsichord.

David Sadler is Second Master in Mathematics and Master in Charge of Statistics at Sydney Grammar School, where he has taught since 1980. He has a BSc from the University of NSW and an MA in Pure Mathematics and a DipEd from Sydney University. In 1979, he taught at Sydney Boys’ High School, and he was a Visiting Fellow at the University of NSW in 1991.

Julia Shea is Head of Mathematics at Newington College, with a BSc and DipEd from the University of Tasmania. She taught for six years at Rosny College, a State Senior College in Hobart, and then for ﬁve years at Sydney Grammar School. She was a member of the Executive Committee of the Mathematics Association of Tasmania for ﬁve years.

Derek Ward has taught Mathematics at Sydney Grammar School since 1991, and is Master in Charge of Database Administration. He has an MSc in Applied Mathematics and a BScDipEd, both from the University of NSW, where he was subsequently Senior Tutor for three years. He has an AMusA in Flute, and sings in the Choir of Christ Church St Laurence.

The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful. The ideas, like the colours or the words, must ﬁt together in a harmonious way. Beauty is the ﬁrst test.

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CHAPTER ONE

Methods in Algebra

Mathematics is the study of structure, pursued using a highly reﬁned form of language in which every word has an exact meaning, and in which the logic is expressed with complete precision. As the structures and the logic of their explanation become more complicated, the language describing them in turn becomes more specialised, and requires systematic study for the meaning to be understood. The symbols and methods of algebra are one aspect of that special language, and ﬂuency in algebra is essential for work in all the various topics of the course.

Study Notes: Several topics in this chapter will probably be quite new — the four cubic identities of Section 1E, solving a set of three simultaneous equations in three variables in Section 1G, and the language of sets in Section 1J. The rest of the chapter is a concise review of algebraic work which would normally have been carefully studied in previous years, and needs will therefore vary as to the amount of work required on these exercises.

1 A

Terms, Factors and Indices

A pronumeral is a symbol that stands for a number. The pronumeral may stand for a known number, or for an unknown number, or it may be a variable, standing for any one of a whole set of possible numbers. Pronumerals, being numbers, can therefore be subjected to all the operations that are possible with numbers, such as addition, subtraction, multiplication and division (except by zero).

Like and Unlike Terms: An algebraic expression is an expression such as

x2+ 2x + 3x2− 4x − 3,

in which pronumerals and numbers and operations are combined. The ﬁve terms in the above expression are x2, 2x, 3x2,−4x and −3. The two like terms x2 and 3x2 can be combined to give 4x2, and the like terms 2x and −4x can be combined to give−2x. This results in three unlike terms 4x2,−2x and −3, which cannot be combined.

WORKEDEXERCISE: x2+ 2x + 3x2− 4x − 3 = 4x2− 2x − 3

Multiplying Terms: To simplify a product like 3xy × (−6x2y) × 12y, it is best to work systematically through the signs, the numerals, and the pronumerals.

WORKEDEXERCISE: (a) 4ab×7bc = 28ab2c (b) 3xy ×(−6x2y)×1

2y = −9x 3

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2 CHAPTER1: Methods in Algebra CAMBRIDGEMATHEMATICS3 UNITYEAR11

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Index Laws: Here are the standard laws for dealing with indices (see Chapter Six for more detail).

1

INDEX LAWS: axay = ax+y (ab)x = axbx ax ay = a x−y a b x = a x bx (ax)n = axn WORKEDEXERCISE: (a) 3x4× 4x3 = 12x7 (b) (48x7y3)÷ (16x5y3) = 3x2 (c) (3a4)3= 27a12 (d) (−5x2)3× (2xy)4 =−125x6× 16x4y4 =−2000x10y4 (e) (6x 4 y)2 3(x2_y3₎3 = 36x8y2 3x6_y9 = 12x 2 y7

Exercise 1A

1. Simplify: (a) 3x − 2y + 5x + 6y (b) 2a2+ 7a − 5a2− 3a (c) 9x2− 7x + 4 − 14x2− 5x − 7 (d) 3a − 4b − 2c + 4a + 2b − c + 2a − b − 2c

2. Find the sum of:

(a) x + y + z, 2x + 3y − 2z and 3x − 4y + z (b) 2a − 3b + c, 15a − 21b − 8c and 24b + 7c + 3a (c) 5ab + bc − 3ca, ab − bc + ca and −ab + 2ca + bc (d) x3− 3x2y + 3xy2, −2x2y − xy2− y3 and x3+ 4y3

3. Subtract:

(a) x from 3x (b) −x from 3x (c) 2a from −4a (d) −b from −5b

4. From: (a) 7x2− 5x + 6 take 5x2− 3x + 2 (b) 4a − 8b + c take a − 3b + 5c (c) 3a + b − c − d take 6a − b + c − 3d (d) ab − bc − cd take −ab + bc − 3cd 5. Subtract: (a) x3− x2+ x + 1 from x3+ x2− x + 1

(b) 3xy2− 3x2y + x3− y3 from x3+ 3x2y + 3xy2+ y3 (c) b3+ c3− 2abc from a3+ b3− 3abc

(d) x4+ 5 + x − 3x3 from 5x4− 8x3− 2x2+ 7 6. Multiply: (a) 5a by 2 (b) 6x by −3 (c) −3a by a (d) −2a2 by −3ab (e) 4x2 by −2x3 (f) −3p2q by 2pq3 7. Simplify:

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CHAPTER1: Methods in Algebra 1B Expanding Brackets 3

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8. If a = −2, ﬁnd the value of: (a) 3a2− a + 4 (b) a4+ 3a3+ 2a2− a

9. If x = 2 and y = −3, ﬁnd the value of: (a) 8x2− y3 (b) x2− 3xy + 2y2

10. Simplify: (a) 5x x (b) −7x3 x (c) −12a2 b −ab (d) −27x6 y7z2 9x3_y3_z 11. Divide: (a) −2x by x (b) 3x3 by x2 (c) x3y2 by x2y (d) a6x3 by −a2x3

(e) 14a5b4 by −2a4b (f) −50a2b5c8 by −10ab3c2 D E V E L O P M E N T 12. Simplify: (a) 3a × 3a × 3a 3a + 3a + 3a (b) 3c × 4c2× 5c3 3c2_{+ 4c}2_{+ 5c}2 (c) ab2× 2b2c3× 3c3a4 a3_b3_{+ 2a}3_b3_{+ 3a}3_b3 13. Simplify: (a) (−2x 2₎3 −4x (b) (3xy3)3 3x2_y4 (c) (−ab)3× (−ab2)2 −a5_b3 (d) (−2a3b2)2× 16a7b (2a2_b)5

14. What must be added to 4x3− 3x2+ 2 to give 3x3+ 7x − 6?

15. Take the sum of 2a − 3b − 4c and −4a + 7b − 5c from the sum of 4c − 2b and 5b − 2a − 2c.

16. If X = 2b + 3c − 5d and Y = 4d − 7c − b, take X − Y from X + Y .

17. Divide the product of (−3x7y5)4 and (−2xy6)3 by (−6x3y8)2.

E X T E N S I O N

18. For what values of x is it true that: (a) x × x ≤ x + x? (b) x × x × x ≤ x + x + x?

1 B

Expanding Brackets

The laws of arithmetic tell us that a(x + y) = ax + ay, whatever the values of a,

x and y. This enables expressions with brackets to be expanded, meaning that

they can be written in a form without brackets.

WORKEDEXERCISE: (a) 3x(x − 2xy) = 3x2− 6x2y (b) a2(a − b) − b2(b − a) = a3− a2b − b3+ ab2 (c) (4x − 2)(4x − 3) = 4x(4x − 3) − 2(4x − 3) = 16x2− 12x − 8x + 6 = 16x2− 20x + 6

Special Quadratic Identities: These three identities are so important that they need to be memorised rather than worked out each time.

2

SQUARE OF A SUM: (A + B)2 = A2+ 2AB + B2 SQUARE OF A DIFFERENCE: (A − B)2 = A2− 2AB + B2 DIFFERENCE OF SQUARES: (A + B)(A − B) = A2− B2

WORKEDEXERCISE:

(a) (4x + 5y)2 = 16x2+ 40xy + 25y2 (square of a sum) (b) t − 1 t 2 = t2− 2 + 1 t2 (square of a diﬀerence) (c) (x2+ 3y)(x2− 3y) = x4− 9y2 (diﬀerence of squares)

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Exercise 1B

1. Expand: (a) 4(a + 2b) (b) x(x − 7) (c) −3(x − 2y) (d) −a(a + 4) (e) 5(a + 3b − 2c) (f) −3(2x − 3y + 5z) (g) −2x(x3− 2x2− 3x + 1) (h) 3xy(2x2y − 5x3)

(i) −2a2b(a2b3− 2a3b)

2. Expand and simplify: (a) 3(x − 2) − 2(x − 5)

(b) −7(2a − 3b + c) − 6(−a + 4b − 2c)

(c) x2(x3− 5x2+ 6x − 1) − 2x(x4+ 10x3− 2x2− 7x + 3) (d) −2x3y(3x2y4− 4xy5+ 5y7)− 3xy2(x2y6+ 2x4y3− 2x3y4)

3. Expand and simplify: (a) (x + 2)(x + 3) (b) (2a + 3)(a + 5) (c) (x − 4)(x + 2) (d) (2b − 7)(b − 3) (e) (3x + 8)(4x − 5) (f) (6− 7x)(5 − 6x)

4. (a) By expanding (A+B)(A+B), prove the special expansion (A+B)2 = A2+2AB +B2. (b) Similarly, prove the special expansions:

(i) (A − B)2= A2− 2AB + B2 (ii) (A − B)(A + B) = A2− B2

5. Expand, using the special expansions: (a) (x − y)2 (b) (a + 3)2 (c) (n − 5)2 (d) (c − 2)(c + 2) (e) (2a + 1)2 (f) (3p − 2)2 (g) (3x + 4y)(3x − 4y) (h) (4y − 5x)2 6. Multiply: (a) a − 2b by a + 2b (b) 2− 5x by 5 + 4x (c) 4x + 7 by itself (d) x2+ 3y by x2− 4y (e) a + b − c by a − b (f) 9x2− 3x + 1 by 3x + 1

7. Expand and simplify: (a) t +1 t 2 (b) t − 1 t 2 (c) t +1 t t − 1 t D E V E L O P M E N T

8. (a) Subtract a(b + c − a) from the sum of b(c + a − b) and c(a + b − c).

(b) Subtract the sum of 2x2− 3(x − 1) and 2x + 3(x2− 2) from the sum of 5x2− (x − 2) and x2− 2(x + 1).

9. Simplify: (a) 14−10− (3x − 7) − 8x (b) 4

a − 2(b − c) −a − (b − 2) 10. Use the special expansions to ﬁnd the value of: (a) 1022 (b) 9992 (c) 203× 197

11. Expand and simplify:

(a) (a − b)(a + b) − a(a − 2b) (b) (x + 2)2− (x + 1)2 (c) (a − 3)2− (a − 3)(a + 3)

(d) (p + q)2− (p − q)2

(e) (2x + 3)(x − 1) − (x − 2)(x + 1) (f) 3(a − 4)(a − 2) − 2(a − 3)(a − 5)

12. If X = x − a and Y = 2x + a, ﬁnd the product of Y − X and X + 3Y in terms of x and a.

13. Expand and simplify: (a) (x − 2)3

(b) (x + y + z)2− 2(xy + yz + zx)

(c) (x + y − z)(x − y + z)

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CHAPTER1: Methods in Algebra 1C Factorisation 5

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14. Prove the identities:

(a) (a + b + c)(ab + bc + ca) − abc = (a + b)(b + c)(c + a)

(b) (ax + by)2+ (ay − bx)2+ c2(x2+ y2) = (x2+ y2)(a2+ b2+ c2)

15. If 2x = a + b + c, show that (x − a)2+ (x − b)2+ (x − c)2+ x2 = a2+ b2+ c2. 16. If (a + b)2+ (b + c)2+ (c + d)2 = 4(ab + bc + cd), prove that a = b = c = d.

1 C

Factorisation

Factorisation is the reverse process of expanding brackets, and will be needed

on a routine basis throughout the course. The various methods of factorisation are listed systematically, but in every situation common factors should always be taken out ﬁrst.

3

METHODS OF FACTORISATION:

HIGHEST COMMON FACTOR: Always try this ﬁrst. DIFFERENCE OF SQUARES: This involves two terms. QUADRATICS: This involves three terms.

GROUPING: This involves four or more terms.

Factoring should continue until each factor is irreducible, meaning that it cannot be factored further.

Factoring by Highest Common Factor and Difference of Squares: In every situation, look for any common factors of all the terms, and then take out the highest common factor.

WORKEDEXERCISE: Factor: (a) 18a2b4− 30b3 (b) 80x4− 5y4 SOLUTION:

(a) The highest common factor of 18a2b4 and 30b3 is 6b3, so 18a2b4− 30b3 = 6b3(3a2b − 5).

(b) 80x4− 5y4= 5(16x4− y4) (highest common factor) = 5(4x2− y2)(4x2+ y2) (diﬀerence of squares)

= 5(2x − y)(2x + y)(4x2+ y2) (diﬀerence of squares again) Factoring Monic Quadratics: A quadratic is called monic if the coeﬃcient of x2 is 1.

Suppose that we want to factor a monic quadratic expression like x2− 13x + 36. We look for two numbers whose sum is −13 (the coeﬃcient of x) and whose product is 36 (the constant).

WORKEDEXERCISE: Factor: (a) x2− 13x + 36 (b) a2+ 12ac − 28c2 SOLUTION:

(a) The numbers with sum −13 and product 36 are−9 and −4, so x2− 13x + 36

= (x − 9)(x − 4).

(b) The numbers with sum 12 and product−28 are 14 and −2, so a2+ 12ac − 28c2

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Factoring Non-monic Quadratics: In a non-monic quadratic like 2x2+ 11x + 12, where the coefficient of x2 is not 1, we look for two numbers whose sum is 11 (the coefficient of x), and whose product is 24 (the product of the constant term and the coefficient of x2).

WORKEDEXERCISE: Factor: (a) 2x2+ 11x + 12 (b) 6s2− 11st − 10t2 SOLUTION:

(a) The numbers with sum 11 and product 24 are 8 and 3, so 2x2+ 11x + 12

= (2x2+ 8x) + (3x + 12) = 2x(x + 4) + 3(x + 4) = (2x + 3)(x + 4).

(b) The numbers with sum−11 and product−60 are −15 and 4, so 6s2− 11st − 10t2

= (6s2− 15st) + (4st − 10t2) = 3s(2s − 5t) + 2t(2s − 5t) = (3s + 2t)(2s − 5t).

Factoring by Grouping: When there are four or more terms, it is sometimes possible to split the expression into groups, factor each group in turn, and then factor the whole expression by taking out a common factor or by some other method.

WORKEDEXERCISE: Factor: (a) 12xy − 9x − 16y + 12 (b) s2− t2+ s − t SOLUTION:

(a) 12xy − 9x − 16y + 12 = 3x(4y − 3) − 4(4y − 3) = (3x − 4)(4y − 3) (b) s2− t2+ s − t = (s + t)(s − t) + (s − t)

= (s − t)(s + t + 1)

Exercise 1C

1. Write as a product of two factors: (a) ax − ay (b) x2+ 3x (c) 3a2− 6ab (d) 12x2+ 18x (e) 6a3+ 2a4+ 4a5 (f) 7x3y − 14x2y2+ 21xy2

2. Factor by grouping in pairs: (a) ax − ay + bx − by (b) a2+ ab + ac + bc (c) x2− 3x − xy + 3y (d) 2ax − bx − 2ay + by (e) ab + ac − b − c (f) 2x3− 6x2− ax + 3a

3. Factor each diﬀerence of squares: (a) x2− 9 (b) 1− a2 (c) 4x2− y2 (d) 25x2− 16 (e) 1− 49k2 (f) 81a2b2− 64

4. Factor each of these quadratic expressions: (a) x2+ 8x + 15 (b) x2− 4x + 3 (c) a2+ 2a − 8 (d) y2− 3y − 28 (e) c2− 12c + 27 (f) p2+ 9p − 36 (g) u2− 16u − 80 (h) x2− 20x + 51 (i) t2+ 23t − 50 (j) x2− 9x − 90 (k) x2− 5xy + 6y2 (l) x2+ 6xy + 8y2 (m) a2− ab − 6b2 (n) p2+ 3pq − 40q2 (o) c2− 24cd + 143d2

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CHAPTER1: Methods in Algebra 1D Algebraic Fractions 7

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5. Write each quadratic expression as a product of two factors: (a) 2x2+ 5x + 2 (b) 3x2+ 8x + 4 (c) 6x2− 11x + 3 (d) 3x2+ 14x − 5 (e) 9x2− 6x − 8 (f) 6x2− 7x − 3 (g) 6x2− 5x + 1 (h) 3x2+ 13x − 30 (i) 12x2− 7x − 12 (j) 12x2+ 31x − 15 (k) 24x2− 50x + 25 (l) 2x2+ xy − y2 (m) 4a2− 8ab + 3b2 (n) 6p2+ 5pq − 4q2 (o) 18u2− 19uv − 12v2

6. Write each expression as a product of three factors: (a) 3a2− 12 (b) x4− y4 (c) x3− x (d) 5x2− 5x − 30 (e) 25y − y3 (f) 16− a4 (g) 4x2+ 14x − 30 (h) x3− 8x2+ 7x (i) x4− 3x2− 4 (j) ax2− a − 2x2+ 2 (k) 16m3− mn2 (l) ax2− a2x − 20a3 D E V E L O P M E N T

7. Factor as fully as possible: (a) 72 + x − x2 (b) (a − b)2− c2 (c) a3− 10a2b + 24ab2 (d) a2− b2− a + b (e) x4− 256 (f) 4p2− (q + r)2 (g) 6x4− x3− 2x2 (h) a2− bc − b + a2c (i) 9x2+ 36x − 45 (j) 4x4− 37x2+ 9 (k) x2y2− 13xy − 48 (l) x(x − y)2− xz2 (m) 20− 9x − 20x2 (n) 4x3− 12x2− x + 3

(o) 12x2− 8xy − 15y2 (p) x2+ 2ax + a2− b2 (q) 9x2− 18x − 315 (r) x4− x2− 2x − 1 (s) 10x3− 13x2y − 9xy2 (t) x2+ 4xy + 4y2− a2+ 2ab − b2 (u) (x + y)2− (x − y)2 E X T E N S I O N 8. Factor fully: (a) a2+ b(b + 1)a + b3 (b) a(b + c − d) − c(a − b + d) (c) (a2− b2)2− (a − b)4 (d) 4x4− 2x3y − 3xy3− 9y4

(e) (x2+ xy)2− (xy + y2)2

(f) (a2− b2− c2)2− 4b2c2

(g) (ax + by)2+ (ay − bx)2+ c2(x2+ y2) (h) x2+ (a − b)xy − aby2

(i) a4+ a2b2+ b4 (j) a4+ 4b4

1 D

Algebraic Fractions

An algebraic fraction is a fraction containing pronumerals. They are manipulated in the same way as arithmetic fractions, and factorisation plays a major role. Addition and Subtraction of Algebraic Fractions: A common denominator is required,

but ﬁnding the lowest common denominator can involve factoring all the denom-inators.

4 ADDITION AND SUBTRACTION OF ALGEBRAIC FRACTIONS: First factor all denominators.

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WORKEDEXERCISE: (a) 1 x − 4− 1 x = x − (x − 4) x(x − 4) = 4 x(x − 4) (b) 2 x2− x− 5 x2− 1= 2 x(x − 1)− 5 (x − 1)(x + 1) = 2(x + 1) − 5x x(x − 1)(x + 1) = 2− 3x x(x − 1)(x + 1)

Multiplication and Division of Algebraic Fractions: The key step here is to factor all numerators and denominators completely before cancelling factors.

5 MULTIPLICATION AND DIVISION OF ALGEBRAIC FRACTIONS: First factor all numerators

and denominators completely. Then cancel common factors. To divide by an algebraic fraction, multiply by its reciprocal in the usual way.

WORKEDEXERCISE: (a) 2a 9− a2 × a − 3 a3_{+ a}= 2a (3− a)(3 + a)× a − 3 a(a2_{+ 1)} =− 2 (a + 3)(a2_{+ 1)} (b) 6abc ab + bc ÷ 6ac a2_{+ 2ac + c}2= 6abc b(a + c) × (a + c)2 6ac = a + c

Simplifying Compound Fractions: A compound fraction is a fraction in which either the numerator or the denominator is itself a fraction.

6 SIMPLIFYING COMPOUND FRACTIONS: Multiply top and bottom by something that

will clear fractions from numerator and denominator together.

WORKEDEXERCISE: (a) 1 2 − 1 3 1 4 + 1 6 = 1 2 − 1 3 1 4 + 1 6 ×12 12 = 6− 4 3 + 2 = 2₅ (b) 1 t + 1 t + 1 1 t − 1 t + 1 = 1 t + 1 t + 1 1 t − 1 t + 1 ×t(t + 1) t(t + 1) = (t + 1) + t (t + 1) − t = 2t + 1

Exercise 1D

1. Simplify: (a) x 2x (b) a a2 (c) 3x 2 9xy (d) 12ab 4a2_b (e) 12xy 2 z 15x2_yz2 (f) uvw 2 u3_v2_w

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CHAPTER1: Methods in Algebra 1D Algebraic Fractions 9

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2. Simplify: (a) x 3 × 3 x (b) a 4 ÷ a 2 (c) x × 3 x2 (d) a 2 2b × b2 a2 (e) 3x 2 4y2 × 2y x (f) x 2 3ay3 ÷ x2 3ay3 (g) 5 a ÷ 10 (h) 2ab 3c × c2 ab2 (i) 8a 3_b 5 ÷ 4ab 15 (j) 2a 3b × 5c2 2a2_b × 3b2 2c (k) 12x 2_yz 8xy3 × 24xy2 36yz2 (l) 3a 2 b 4b3_c × 2c2 8a3 ÷ 6ac 16b2

3. Write as a single fraction: (a) x 2 + x 5 (b) a 3 − a 6 (c) x 8 − y 12 (d) 2a 3 + 3a 2 (e) 7b 10− 19b 30 (f) xy 30 − xy 18 (g) 1 x + 1 2x (h) 3 4x+ 4 3x (i) 1 a− 1 b (j) x + 1 x (k) a + b a (l) 1 2x− 1 x2 4. Simplify: (a) x + 1 2 + x + 2 3 (b) 2x − 1 5 − x + 3 2 (c) 2x + 1 3 − x − 5 6 + x + 4 4 (d) 3x − 7 5 + 4x + 3 2 − 2x − 5 10 (e) x − 5 3x − x − 3 5x (f) 1 x − 1 x + 1 (g) 1 x + 1− 1 x + 1 (h) 2 x − 3+ 3 x − 2 (i) 2 x + 3− 2 x − 2 (j) x x + y + y x − y (k) a x + a − b x + b (l) x x − 1− x x + 1

5. Factor where possible and then simplify: (a) a ax + ay (b) 3a 2_{− 6ab} 2a2_{b − 4ab}2 (c) x 2 + 2x x2− 4 (d) a 2_{− 9} a2_{+ a − 12} (e) x 2 + 2xy + y2 x2− y2 (f) x 2_{+ 10x + 25} x2_{+ 9x + 20} (g) ac + ad + bc + bd a2_{+ ab} (h) y 2_{− 8y + 15} 2y2− 5y − 3

(i) 9ax + 6bx − 6ay − 4by 9x2_{− 4y}2 6. Simplify: (a) 3x + 3 2x × x2 x2− 1 (b) a 2_{+ a − 2} a + 2 × a2_{− 3a} a2_{− 4a + 3} (c) c 2 + 5c + 6 c2− 16 ÷ c + 3 c − 4 (d) x 2_{− x − 20} x2− 25 × x2− x − 2 x2_{+ 2x − 8}÷ x + 1 x2_{+ 5x} (e) ax + bx − 2a − 2b 3x2_{− 5x − 2} × 9x2_{− 1} a2_{+ 2ab + b}2 (f) 2x 2 + x − 15 x2_{+ 3x − 28}÷ x2+ 6x + 9 x2− 4x ÷ 6x2− 15x x2− 49 7. Simplify: (a) 1 x2_{+ x}+ 1 x2− x (b) 1 x2_{− 4}+ 1 x2_{− 4x + 4} (c) 1 x − y + 2x − y x2_{− y}2 (d) 3 x2_{+ 2x − 8}− 2 x2_{+ x − 6} (e) x a2_{− b}2 − x a2_{+ ab} (f) 1 x2_{− 4x + 3}+ 1 x2_{− 5x + 6}− 1 x2_{− 3x + 2}

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8. Simplify: (a) b − a a − b (b) v 2_{− u}2 u − v (c) x 2_{− 5x + 6} 2− x (d) 1 a − b− 1 b − a (e) m m − n + n n − m (f) x − y y2_{+ xy − 2x}2 D E V E L O P M E N T

9. Study the worked exercise on compound fractions and then simplify: (a) 1− 1 2 1 +1₂ (b) 2 + 1 3 5−2₃ (c) 1 2 − 1 5 1 +₁₀1 (d) 17 20 − 3 4 4 5 − 3 10 (e) 1 x 1 + 2 x (f) t − 1 t t + 1_t (g) ₁ 1 b + 1 a (h) x y + y x x y −yx (i) 1− 1 x+1 1 x + 1 x+1 (j) 3 x+2 − 2 x+1 5 x+2 − 4 x+1 10. If x = 1 λ and y = 1 1− x and z = y y − 1, show that z = λ. 11. Simplify: (a) x 4_{− y}4 x2− 2xy + y2 ÷ x2+ y2 x − y (b) 8x2+ 14x + 3 8x2− 10x + 3 × 12x2− 6x 4x2_{+ 5x + 1} ÷ 18x2− 6x 4x2_{+ x − 3} (c) (a − b) 2_{− c}2 ab − b2− bc × c a2_{+ ab − ac} ÷ ac − bc + c2 a2− (b − c)2 (d) x − y x + x3_{+ y}3 xy2 − x2_{+ y}2 x2 (e) x + 4 x − 4− x − 4 x + 4 (f) 4y x2_{+ 2xy} − 3x xy + 2y2 + 3x − 2y xy (g) 8x x2_{+ 5x + 6}− 5x x2_{+ 3x + 2}− 3x x2_{+ 4x + 3} (h) 1 x − 1+ 2 x + 1− 3x − 2 x2− 1− 1 x2_{+ 2x + 1} 12. (a) Expand x + 1 x 2 . (b) Suppose that x +1

x = 3. Use part (a) to evaluate x

2₊ 1

x2 without attempting to ﬁnd the value of x.

13. Simplify these algebraic fractions:

(a) 1 (a − b)(a − c) + 1 (b − c)(b − a)+ 1 (c − a)(c − b) (b) 1 + 45 x − 8− 26 x − 6 3− 65 x + 7+ 8 x − 2 (c) 2− 3n m + 9n2− 2m2 m2_{+ 2mn} ÷ ⎛ ⎜ ⎜ ⎝_m1 − 1 m − 2n − 4n 2 m + n ⎞ ⎟ ⎟ ⎠ (d) 1 x + 1 x + 2 × 1 x + 1 x − 2 ÷ x − 4 x x2− 2 + 1 x2

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CHAPTER1: Methods in Algebra 1E Four Cubic Identities 11

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1 E

Four Cubic Identities

The three special quadratic identities will be generalised later to any degree. For now, here are the cubic versions of them. They will be new to most people.

7

CUBE OF A SUM: (A + B)3 = A3+ 3A2B + 3AB2+ B3 CUBE OF A DIFFERENCE: (A − B)3 = A3− 3A2B + 3AB2− B3

DIFFERENCE OF CUBES: A3− B3 = (A − B)(A2+ AB + B2) SUM OF CUBES: A3+ B3 = (A + B)(A2− AB + B2)

The proofs of these identities are left to the ﬁrst two questions in the following exercise.

WORKEDEXERCISE: Here is an example of each identity. (a) (x + 5)3= x3+ 15x2+ 75x + 125

(b) (2x − 3y)3= 8x3− 36x2y + 54xy2− 27y3

(c) x3− 8= (x − 2)(x2+ 2x + 4)

(d) 43+ 53 = (4 + 5)(16− 20 + 25)= 9 × 21 = 33× 7

WORKEDEXERCISE: (a) Simplify a 3_{+ 1} a + 1. (b) Factor a 3_{− b}3 + a − b. SOLUTION: (a) a 3_{+ 1} a + 1 = (a + 1)(a2_{− a + 1)} a + 1 = a2− a + 1 (b) a3− b3+ a − b = (a − b)(a2+ ab + b2) + (a − b) = (a − b)(a2+ ab + b2+ 1)

Exercise 1E

1. (a) Prove the factorisation A3− B3 = (A − B)(A2+ AB + B2) by expanding the RHS. (b) Similarly, prove the factorisation A3+ B3 = (A + B)(A2− AB + B2).

2. (a) Prove the identity (A + B)3 = A3+ 3A2B + 3AB2+ B3 by writing (A + B)3 = (A + B)(A2+ 2AB + B2) and expanding.

(b) Similarly, prove the identity (A − B)3 = A3− 3A2B + 3AB2− B3.

3. Expand: (a) (a + b)3 (b) (x − y)3 (c) (b − 1)3 (d) (p + 2)3 (e) (1− c)3 (f) (t − 3)3 (g) (2x + 5y)3 (h) (3a − 4b)3 4. Factor: (a) x3+ y3 (b) a3− b3 (c) y3+ 1 (d) g3− 1 (e) b3− 8 (f) 8c3+ 1 (g) 27− t3 (h) 125 + a3 (i) 27h3− 1 (j) u3− 64v3 (k) a3b3c3+ 1000 (l) 216x3+ 125y3

5. Write as a product of three factors: (a) 2x3+ 16 (b) a4− ab3 (c) 24t3+ 81 (d) x3y − 125y (e) 250p3− 432q3 (f) 27x4+ 1000xy3 (g) 5x3y3− 5 (h) x6+ x3y3

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6. Simplify: (a) x 3_{− 1} x2− 1 (b) a 2_{− 3a − 10} a3_{+ 8} (c) a 3_{+ 1} 6a2 × 3a a2_{+ a} (d) x 2_{− 9} x4_{− 27x}÷ x + 3 x2_{+ 3x + 9} 7. Simplify: (a) 3 a − 2 − 3a a2_{+ 2a + 4} (b) 1 x3− 1+ x + 1 x2_{+ x + 1} (c) 1 x2− 2x − 8− 1 x3_{+ 8} (d) a 2 a3_{+ b}3 + a − b a2− ab + b2 + 1 a + b D E V E L O P M E N T

8. Factor as fully as possible: (a) a3+ b3+ a + b (b) x6− 64 (c) 2a4− 3a3+ 16a − 24 (d) (x + y)3− (x − y)3 (e) s3− t3+ s2− t2 (f) (t − 2)3+ (t + 2)3 (g) (a − 2b)3+ (2a − b)3 (h) x6− 7x3− 8 (i) u7+ u6+ u + 1 (j) 2 + x3− 3x6 (k) x7− x3+ 8x4− 8 (l) a5+ a4+ a3+ a2+ a + 1 9. Simplify: (a) 6a 2_{+ 6} a2_{+ a + 1}× a3− 1 a3− 3a2 × a3+ a2 a4− 1 (b) x 4_{− 8x} x2− 4x − 5× x2+ 2x + 1 x3− x2− 2x÷ x2+ 2x + 4 x − 5 (c) (a + 1) 3_{− (a − 1)}3 3a3_{+ a} (d) 1 x − 3− 8x x3− 27− x − 3 x2_{+ 3x + 9} (e) 3x 2_{+ 2x + 4} x3− 1 − x + 1 x2_{+ x + 1}− 2 x − 1 (f) 1 + x + x 2 1− x3 + x − x2 (1− x)3 E X T E N S I O N

10. Find the four quartic identities that correspond to the cubic identities in this exercise. That is, ﬁnd the expansions of (A + B)4 and (A − B)4 and ﬁnd factorisations of A4+ B4 and A4− B4.

11. Factor as fully as possible: (a) x7+ x (b) x12 − y12 12. If x + y = 1 and x3+ y3 = 19, ﬁnd the value of x2+ y2.

13. Simplify (x − y)3+ (x + y)3+ 3(x − y)2(x + y) + 3(x + y)2(x − y).

14. If a + b + c = 0, show that (2a − b)3+ (2b − c)3+ (2c − a)3= 3(2a − b)(2b − c)(2c − a). 15. Simplify a 4_{− b}4 a2− 2ab + b2 ÷ a2b + b3 a3− b3 × a2b − ab2+ b3 a4_{+ a}2_b2_{+ b}4. 16. Simplify (1 + a)2÷ ⎛ ⎜ ⎜ ⎝1 + a 1− a + a 1 + a + a2 ⎞ ⎟ ⎟ ⎠.

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CHAPTER1: Methods in Algebra 1F Linear Equations and Inequations 13

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1 F

Linear Equations and Inequations

The rules for solving equations and for solving inequations are the same, except for a qualiﬁcation about multiplying or dividing an inequation by a negative:

8

LINEAR EQUATIONS: Any number can be added to or subtracted from both sides. Both sides can be multiplied or divided by any nonzero number.

LINEAR INEQUATIONS: The rules for inequations are the same as those for equations, except that when both sides are multiplied or divided by a negative number, the inequality sign is reversed.

WORKEDEXERCISE: Solve: (a) 4− 7x

4x − 7 = 1 (b) x − 12 < 5 + 3x SOLUTION: (a) 4− 7x 4x − 7 = 1 × (4x − 7) 4 − 7x = 4x − 7 + 7x 4 = 11x − 7 + 7 11 = 11x ÷ 11 x = 1 (b) x − 12 < 5 + 3x − 3x −2x − 12 < 5 + 12 −2x < 17 ÷ (−2) x > −812

Because of the division by the neg-ative, the inequality was reversed. Changing the Subject of a Formula: Similar sequences of operations allow the subject

of a formula to be changed from one pronumeral to another.

WORKEDEXERCISE: Given the formula y = x + 1

x + a:

(a) change the subject to a, (b) change the subject to x. SOLUTION: (a) y = x + 1 x + a × (x + a) xy + ay = x + 1 − xy ay = x + 1 − xy ÷ y a = x + 1 − xy y (b) y = x + 1 x + a × (x + a) xy + ay = x + 1 xy − x = 1 − ay x(y − 1) = 1 − ay ÷ (y − 1) x = 1− ay y − 1

Exercise 1F

1. Solve: (a) −2x = −20 (b) 3x > 2 (c) −a = 5 (d) x −4 ≤ −1 (e) −1 − x = 0 (f) 0·1y = 5 (g) 2t < t (h) −1₂x = 8 2. Solve: (a) 3x − 5 = 22 (b) 4x + 7 ≥ −13 (c) 1− 2x < 9 (d) 6x = 3x − 21 (e) −13 ≤ 5a − 6 (f) −2 > 4 + t 5 (g) 19 = 3− 7y (h) 23−u 3 ≥ 7

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3. Solve: (a) 5x − 2 < 2x + 10 (b) 5− x = 27 + x (c) 16 + 9a > 10 − 3a (d) 13y − 21 ≤ 20y − 35 (e) 13− 12x ≥ 6 − 3x (f) 3(x + 7) = −2(x − 9) (g) 8 + 4(2− x) > 3 − 2(5 − x) (h) 7x − (3x + 11) = 6 − (15 − 9x) (i) 4(x + 2) = 4x + 9 (j) 3(x − 1) < 2(x + 1) + x (k) (x − 3)(x + 6) ≤ (x − 4)(x − 5) (l) (1 + 2x)(4 + 3x) = (2 − x)(5 − 6x) (m) (x + 3)2 > (x − 1)2 (n) (2x − 5)(2x + 5) = (2x − 3)2 4. Solve: (a) x 8 = 1 2 (b) a 12 = 2 3 (c) y 20 < 4 5 (d) 1 x = 3 (e) 2 a = 5 (f) 3 = 9 2y (g) 2x + 1 5 ≥ −3 (h) 5a 3 − 1 ≥ 3a 5 + 1 (i) 7− 4x 6 < 1 (j) 5 + a a =−3 (k) 9− 2t t = 13 (l) 6−c 3 > c (m) 1 a+ 4 = 1− 2 a (n) 4 x − 1 =−5 (o) 3x 1− 2x = 7 (p) 11t 8t + 13 =−2 5. Solve: (a) x 3 − x 5 ≥ 2 (b) a 10 − a 6 < 1 (c) x 6 + 2 3 = x 2 − 5 6 (d) 1 x − 3 = 1 2x (e) 1 2x− 2 3 = 1− 1 3x (f) x 3 − 2 < x 2 − 3 (g) x − 2 3 > x + 4 4 (h) 3 x − 2 = 4 2x + 5 (i) x + 1 x + 2 = x − 3 x + 1 (j) (3x − 2)(3x + 2) (3x − 1)2 = 1 (k) a + 5 2 − a − 1 3 > 1 (l) 3 4 − x + 1 12 ≤ 2 3 − x − 1 6 (m) 2x 5 + 2− 3x 4 < 3 10− 3− 5x 2 (n) 3₄(x − 1) −12(3x + 2) = 0 (o) 4x + 1 6 − 2x − 1 15 = 3x − 5 5 − 6x + 1 10 (p) 7(1− x) 12 − 3 + 2x 9 ≥ 5(2 + x) 6 − 4− 5x 18

6. (a) If v = u + at, ﬁnd a when t = 4, v = 20 and u = 8.

(b) Given that v2 = u2+ 2as, ﬁnd the value of s when u = 6, v = 10 and a = 2. (c) Suppose that 1

u +

1

v =

1

t. Find v, given that u = −1 and t = 2. (d) If S = −15, n = 10 and a = −24, ﬁnd , given that S = n₂(a + ).

(e) Temperatures in degrees Fahrenheit and degrees Celsius are related by the formula

F = 95C + 32. Find the value of C that corresponds to F = 95.

(f) Suppose that the variables c and d are related by the formula 3

c + 1 =

5

d − 1. Find c

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CHAPTER1: Methods in Algebra 1F Linear Equations and Inequations 15

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D E V E L O P M E N T

7. Solve each of the following inequations for the given domain of the variable, and graph each solution on the real number line:

(a) 2x − 3 < 5, where x is a positive integer. (b) 1− 3x ≤ 16, where x is a negative integer.

(c) 4x + 5 > 2x − 3, where x is a real number. (d) 7− 2x ≥ x + 1, where x is a real number.

(e) 4≤ 2x < 14, where x is an integer. (f) −12 < 3x < 9, where x is an integer. (g) 1 < 2x + 1 ≤ 11, where x is a real number. (h) −10 ≤ 2 − 3x ≤ −1, where x is a real number.

8. Solve each of these problems by constructing and then solving a linear equation:

(a) Five more than twice a certain number is one more than the number itself. What is the number?

(b) I have $175 in my wallet, consisting of $10 and $5 notes. If I have twice as many $10 notes as $5 notes, how many $5 notes do I have?

(c) My father is 24 years older than me, and 12 years ago he was double my age. How old am I now?

(d) The fuel tank in my new car was 40% full. I added 28 litres and then found that it was 75% full. How much fuel does the tank hold?

(e) A certain tank has an inlet valve and an outlet valve. The tank can be ﬁlled via the inlet valve in 6 minutes and emptied (from full) via the outlet valve in 10 minutes. If both valves are operating, how long would it take to ﬁll the tank if it was empty to begin with?

(f) A basketball player has scored 312 points in 15 games. How many points must he average per game in his next 3 games to take his overall average to 20 points per game? (g) A cyclist rides for 5 hours at a certain speed and then for 4 hours at a speed 6 km/h greater than her original speed. If she rides 294 km altogether, what was her initial speed?

(h) Two trains travel at speeds of 72 km/h and 48 km/h respectively. If they start at the same time and travel towards each other from two places 600 km apart, how long will it be before they meet?

9. Rearrange each formula so that the pronumeral written in the brackets is the subject: (a) a = bc − d [b] (b) t = a + (n − 1)d [n] (c) p q + r = t [r] (d) u = 1 +3 v [v] (e) a 2 − b 3 = a [a] (f) 1 f + 2 g = 5 h [g] (g) x = y y + 2 [y] (h) a = b + 5 b − 4 [b] (i) c = 7 + 2d 5− 3d [d] (j) u = v + w − 1 v − w + 1 [v] 10. Solve: (a) x x − 2+ 3 x − 4 = 1 (b) 3a − 2 2a − 3− a + 17 a + 10 = 1 2 E X T E N S I O N 11. (a) Show that x − 1

x − 3 = 1 + 2 x − 3. (b) Hence solve x − 1 x − 3− x − 3 x − 5 = x − 5 x − 7− x − 7 x − 9.