Mathematics Grade 10 Study Guide 1/2

Study Guide 1/2

10

Mathematics

CAPS

Mathematics FET Phase Grade 10 • Study Guide 1/2 home classroom college workplace

2210-E-MAM-SG01

9 781990

948398

Mathematics Study Guide 1/2 – Grade 10

2210-E-MAM-SG01

9 781990

948398

CAPS-aligned

Prof. C Vermeulen, Lead author P de Swardt H Otto M Sherman E van Heerden L Young

1.5.4 Quadratic trinomial....................................................................................22

TABLE OF CONTENTS

Exercise 1.5.2: Quadratic trinomials and mixed factorisation............... 24

1.5.5 Sum and difference of cubes....................................................................25 PREFACE..................................................................................................................... 1 THEME 1: NUMBERS AND ALGEBRAIC EXPRESSIONS........................................ 7 1.1

Exercise 1.5.3: Sum and difference of cubes.................................................. 27 Self-evaluation............................................................................................................ 27

Introduction....................................................................................................................7

1.6

Algebraic expressions involving fractions .........................................28

Exercise 1.1: The number system....................................................................... 10

1.6.2 Multiplication and division of algebraic fractions...........................30

Rounding of real numbers........................................................................11

Exercise 1.6.3: Finding the LCM..........................................................................33

The number system...................................................................................... 7

1.6.1 Simplifying single algebraic fractions using factors........................28

Summary of the real number system...................................................... 7

Exercise 1.6.1: Simplifying algebraic fractions using factors.................. 30

1.2

Between which two integers does a surd fall?...................................10

1.6.3 Finding the lowest common multiple (LCM)......................................32

1.3

Exercise 1.2: Estimating surds............................................................................. 11 Self-evaluation............................................................................................................ 12

Exercise 1.6.4: Adding and subtracting algebraic fractions.................... 35

Exercise 1.6.2: Multiplication and division of algebraic fractions........ 31

Exercise 1.3: Rounding numbers........................................................................ 12

1.6.4 Adding and subtracting algebraic fractions.......................................33

Multiplication of algebraic expressions..............................................13

1.6.5 Simplifying expressions involving fractions within fractions......36

1.4.1 Algebraic expressions................................................................................13

Exercise 1.6.5: Simplifying expressions involving fractions within fractions.......................................................................................................... 38

1.4

1.4.2 Products.........................................................................................................13

Self-evaluation............................................................................................................ 38

Exercise 1.4: Multiplication of algebraic expressions................................ 15

Factorising algebraic expressions.........................................................17

Summary of theme.................................................................................................... 39

1.5.1 Common factor.............................................................................................18

THEME 2: EXPONENTS............................................................................................40

1.5.2 Difference of two squares.........................................................................19

Introduction................................................................................................................. 40

1.5

Self-evaluation............................................................................................................ 17

End of theme exercise.............................................................................................. 39 Revision exercise........................................................................................................ 41

1.5.3 Grouping terms (new work) ...................................................................20 Exercise 1.5.1: Common factor, difference of squares and grouping...... 21 © Optimi

i

G10 – Mathematics

G10 – Mathematics – Study Guide 1/2 2.1

Composite bases .........................................................................................42

3.3

Exercise 2.1: Composite bases............................................................................. 44

The formula for the general term of a linear sequence..................61 Exercise 3.3: The formula for the general term of a linear sequence.. 63

Self-evaluation............................................................................................................ 44

3.4

Geometric patterns.....................................................................................64

Rational exponents.....................................................................................47

3.5

Exercise 3.4: Geometric patterns........................................................................ 66

2.2

Simplifying by factorising.........................................................................45

2.3

Exercise 2.2: Simplifying by factorising........................................................... 46 Exercise 2.3: Rational exponents........................................................................ 47

Self-evaluation............................................................................................................ 63

Problem-solving involving repeated patterns...................................67

Exercise 3.5: Problem-solving involving repeated patterns.................... 68

Self-evaluation............................................................................................................ 48

Self-evaluation............................................................................................................ 68

2.4 Equations.......................................................................................................48

Summary of theme.................................................................................................... 69

2.4.1 Equations with the unknown in the base............................................48

End of theme exercise.............................................................................................. 69

2.4.2 Equations with the unknown in the exponent ..................................49

THEME 4: EQUATIONS AND INEQUALITIES........................................................... 71

Exercise 2.4: Equations........................................................................................... 50

Introduction................................................................................................................. 71

Self-evaluation............................................................................................................ 52

Revision exercise........................................................................................................ 71

Summary of theme.................................................................................................... 52

4.1

Linear equations..........................................................................................74

THEME 3: NUMBER PATTERNS..............................................................................54

4.2

Exercise 4.1: Linear equations............................................................................. 75

End of theme exercise.............................................................................................. 52

3.1

3.2

Introduction................................................................................................................. 54 Revision exercise........................................................................................................ 54

The general term of a number pattern................................................56

4.3

Introduction................................................................................................................. 56 Exercise 3.1: The general term of a number pattern.................................. 59

Using the general term..............................................................................59

4.4

Exercise 3.2: Using the general term................................................................. 60

Quadratic equations...................................................................................76

Exercise 4.2: Quadratic equations...................................................................... 78 Self-evaluation............................................................................................................ 79

Linear equation systems...........................................................................80

Exercise 4.3: Linear equation systems............................................................. 83 Self-evaluation............................................................................................................ 84

Literal equatons (changing the subject of a formula).....................85

Exercise 4.4: Literal equations............................................................................. 86

ii

4.5

Linear inequalities......................................................................................86

The trigonometric ratios of 0° and 90°......................................................... 110

Exercise 4.5: Linear inequalities......................................................................... 88 Self-evaluation............................................................................................................ 89 Summary of theme.................................................................................................... 89 End of theme exercise.............................................................................................. 89

THEME 5: TRIGONOMETRY..................................................................................... 91 5.1

5.2

5.3

5.4

Introduction................................................................................................................. 91

The definitions of the trigonometric ratios sin θ, cos θ and tan θ in right-angled triangles................................................................91

5.6

Finding the values of trigonometrical ratios using a calculator ....113

5.7

Exercise 5.6: Finding the values of trigonometric ratios using a calculator.................................................................................................... 114

Solving trigonometric equations (calculating sizes of angles) .115

5.8

Exercise 5.7: Solving trigonometric equations for acute angles......... 116

Solving right-angled triangles..............................................................116

Exercise 5.8 Solving right-angled triangles................................................. 118

Exercise 5.2: Inverse trigonometric ratios in right-angled triangles... 96

End of theme exercise .......................................................................................... 121

Self-evaluation......................................................................................................... 120

Definitions of the inverse trigonometric ratios, cosec θ, sec θ and cot θ, in right-angled triangles.......................................................94

Summary of theme................................................................................................. 121

Self-evaluation............................................................................................................ 97

THEME 6: FUNCTIONS...........................................................................................124

The definitions of the trigonometric ratios sin θ, cos θ and tan θ and their inverses in the Cartesian plane................................98

Exercise 5.3: Applying trigonometric ratios in the Cartesian plane.100

Using the theorem of Pythagoras and trigonometric ratios...... 103

Introduction.............................................................................................................. 124

6.1

Functions and relations.........................................................................125

6.2

Exercise 6.1: Functions and relations............................................................ 127

Linear functions........................................................................................128

6.2.1 Investigating the properties of linear functions............................ 128

Exercise 5.4: Using the theorem of Pythagoras and trigonometric ratios.............................................................................................. 106

Development exercise........................................................................................... 128 Summary of the properties of linear functions.......................................... 132

Self-evaluation......................................................................................................... 109

Direction and steepness of the graph............................................................. 132

Finding the trigonometric ratios of the special angles without using a calculator ....................................................................................109

Intercepts with the axes....................................................................................... 133

The trigonometric ratios of 45° ....................................................................... 109

Domain and range.................................................................................................. 133

The trigonometric ratios of 60° and 30°....................................................... 110 © Optimi

Self-evaluation......................................................................................................... 112

Exercise 5.1: Basic trigonometric ratios in right-angled triangles ...... 93

Revision exercise..................................................................................................... 103

5.5

Exercise 5.5: Working with special angles without using a calculator....112

iii

G10 – Mathematics

G10 – Mathematics – Study Guide 1/2 6.2.2 Function notation.....................................................................................135

6.4

Hyperbolic functions...............................................................................158

6.2.3 Parallel and perpendicular lines........................................................135

6.4.1 Investigating the properties of hyperbolic functions................... 159

6.2.4 Drawing sketch graphs of linear functions...................................... 136

Development exercise........................................................................................... 159

Exercise 6.2.1: Drawing straight line graphs.............................................. 138

6.4.2 The influence of ​a​ on the graph of a hyperbola.............................. 162

6.2.6 Intersecting lines......................................................................................142

6.4.5 Domain and range....................................................................................163

Exercise 6.2.3: Intersecting lines.................................................................... 144

6.4.6 Drawing sketch graphs of hyperbolas............................................... 163

Quadratic functions.................................................................................146

6.4.8 Restrictions on hyperbolas...................................................................166

6.3.1 Investigating the properties of quadratic functions..................... 146

Exercise 6.4.1: Hyperbolas.................................................................................. 167

6.2.5 Finding the equation of a straight line graph................................. 140

6.4.3 The influence of ​q​ on the graph of a hyperbola.............................. 163

Exercise 6.2.2: Finding the equation of a straight line graph.............. 142

6.4.4 Asymptotes.................................................................................................163

Self-evaluation......................................................................................................... 145

6.4.7 Finding the equation of a hyperbola.................................................. 165

Development exercise........................................................................................... 146

Self-evaluation......................................................................................................... 168

6.3

6.3.2 The influence of ​a​on the graph of a parabola ............................... 149

6.5

6.3.3 The turning point of a parabola..........................................................150

6.5.1 Investigating the properties of exponential functions................ 169

6.3.4 The influence of ​q​ on the graph of a parabola................................ 150

Development exercise........................................................................................... 169

6.3.5 Domain and range....................................................................................151

Exponential functions.............................................................................168

6.5.2 The influence of ​a​in ​y = ​a​ 𝑥​​ on the graph of an exponential function...............................................................................172

6.3.6 Axis of symmetry......................................................................................151 6.3.8 Drawing sketch graphs of parabolas.................................................. 152

6.5.3 The influence of ​b​in ​y = b . ​a​ 𝑥​on the graph of an exponential function.........................................................................173

Exercise 6.3.1: Drawing sketch graphs of parabolas............................... 154

6.5.5 Domain and range....................................................................................173

6.3.7 Intercepts with the axes.........................................................................151

6.5.4 The influence of ​q ​in ​y = ​a​ 𝑥​ + q​on the graph of an exponential function.........................................................................173

6.3.9 Finding the equation of a parabola ................................................... 154

Exercise 6.3.2: Finding the equation of a parabola.................................. 157

6.5.6 Reflections..................................................................................................174

Self-evaluation......................................................................................................... 158

Exercise 6.5.1: Properties of exponential graphs...................................... 174

6.5.7 Drawing sketch graphs of exponential functions.......................... 175

iv

Exercise 6.5.2: Drawing sketch graphs of exponential functions....... 176

6.5.8 Finding the equation of an exponential graph............................... 176

THEME 8: EUCLIDEAN GEOMETRY......................................................................212

Exercise 6.5.3: Finding the equation of an exponential function....... 177

Introduction.............................................................................................................. 212

6.6

8.1

Self-evaluation......................................................................................................... 179

Summary: The effect of ​a​and q on each type of graph................. 179

Lines............................................................................................................................. 215 Revision exercise: Basic geometry: lines and angles............................... 217

Summary of theme................................................................................................. 181

Triangles..................................................................................................................... 218

End of theme exercise........................................................................................... 182

Properties of triangles.......................................................................................... 219

THEME 7: TRIGONOMETRIC FUNCTIONS............................................................186

7.1

Triangles: Similarity.............................................................................................. 219

Introduction.............................................................................................................. 186

Triangles: Congruence.......................................................................................... 221

Revision exercise..................................................................................................... 186

Drawing accurate graphs of the sin, cos and tan function.......... 189

Exercise 7.1.1: Drawing accurate graphs of sine function y = sin 𝑥 . 189

8.2

Exercise 7.1.2: Drawing accurate graphs of the cosine function y = cos 𝑥....................................................................................................................... 192

7.2

7.3

Basic geometry: Lines, angles and triangles................................... 213

Exercise 7.1.3: Drawing accurate graphs of the tangent function y = tan 𝑥....................................................................................................................... 194

8.3

Self-evaluation......................................................................................................... 197

Exercise 8.1: Basic geometry: lines, angles and triangles..................... 222 Self-evaluation......................................................................................................... 226

The mid-point theorem..........................................................................226

Exercise 8.2: The mid-point theorem............................................................. 229 Self-evaluation......................................................................................................... 232

Special quadrilaterals.............................................................................232

Drawing and interpreting sketch graphs of trigonometric functions.........................................................................198

Exercise 8.3: Special quadrilaterals................................................................ 239

Exercise 7.2: Drawing and interpreting sketch graphs of the trigonometric functions....................................................................................... 203

End of theme exercise........................................................................................... 243

Self-evaluation......................................................................................................... 242 Summary of theme................................................................................................. 243

Finding the equations of trigonometric graphs and interpreting graphs.................................................................................204

PAPERS...................................................................................................................245

Exercise 7.3: Finding the equations of trigo­nometric graphs and interpreting graphs................................................................................................ 206

EUCLIDIAN GEOMETRY: ACCEPTED REASONS..................................................254

FORMULAE..............................................................................................................251

Self-evaluation......................................................................................................... 209

INDEX......................................................................................................................257

Summary of theme................................................................................................. 210 © Optimi

End of theme exercise........................................................................................... 210

v

G10 – Mathematics

PREFACE

Term

In Grade 10, mathematics is an optional subject (as an alternative to mathematical literacy) for the first time. There may be various reasons why you might choose mathematics as a subject, for example to prepare you for a field of study where Grade 12 mathematics is a prerequisite, or a career in which a background in mathematics would be advantageous.

1

(11 weeks)

In general, mathematics in Grade 10 to 12 involves more abstract concepts and more complex procedures than in Grade 1 to 9. Mastering mathematics in Grade 10 to 12 requires more time, commitment, critical thought and reflection than in Grade 1 to 9.

2

This product consists of two study guides and two facilitator’s guides, which are based on the concepts of Optimi’s GuidEd Learning™ model to help you achieve success in your study of mathematics. These books cover all work required for Grade 10 mathematics and have been compiled in accordance with the CAPS guidelines as required by the Department of Basic Education.

(11 weeks)

Number of weeks 3

1. Algebraic expressions

2

3. Number patterns

2 2 3 4 1 3 3 2

The study guides are supported by supplementary lesson structures on the Optimi Learning Platform (OLP), which is an online platform. These lesson structures offer continuous guidance to support and enrich your learning process. This guidance is based on the latest insights in education, cognitive psychology and neuroscience. Note that the study guides can also be used independently of the OLP.

3

(10 weeks)

The study guides and facilitator’s guides are divided into 15 themes. Study guide 1/2 and facilitator’s guide 1/2 cover themes 1 to 8 (terms 1 and 2) and study guide 2/2 and facilitator’s guide 2/2 cover themes 9 to 15 (terms 3 and 4).

The themes correspond with the CAPS guidelines with regard to content and time allocation and represent the year plan.

2 2 2 1 1

4

(10 weeks)

Theme

2 4 4

2. Exponents

4. Equations and inequalities 5. Trigonometry 6. Functions

7. Trigonometric functions 8. Euclidean geometry

Assessment* Investigation or project Test

Assignment or test

Practice papers are included in the textbook.

June examination

10. Finance and growth

Test

9. Analytical geometry 11. Statistics

12. Problems in two dimensions 13. Euclidean geometry 14. Measurement 15. Probability Revision

Practice papers are included in the textbook.

Test Test End-of-year examination

* You will find the latest and most comprehensive information on assessment in the portfolio book and assessment plan. © Optimi

1

G10 – Mathematics

G10 – Mathematics – Study Guide 1/2

Time allocation

Each theme has the same structure in order to make it easier for you to navigate through them. The structure is as follows:

According to the CAPS requirements, at least 4,5 hours should be spent on teaching mathematics per week. For example, 13,5 hours (three weeks × 4,5 hours per week) will be spent on teaching Theme 1 (algebraic expressions).

Themes have not been sub-divided into lessons; you and your facilitator are at liberty to complete as much content per session and per week as your progress allows. If you work at a slower pace, the necessary adaptations should be done so that you will still be able to master all the work in time.

Introduction

What this theme is about This briefly tells you what the theme is about without providing details or using “difficult” or unknown concepts. A comprehensive list of the learning outcomes you need to master in a specific theme is given as a summary at the end of the theme.

Note that the teaching time referred to above does not include the time during which you should apply and practise the knowledge and concepts you have learned. For this purpose, various exercises are provided throughout each theme. These exercises involve different ways of applying and practising new knowledge and cover various degrees of difficulty. You should try to do all of these exercises. Complete solutions are provided in the facilitator’s guide.

Previous knowledge

This section tells you what existing knowledge you need to master the theme involved.

The learning activities available in the OLP’s lesson structures involve different formats and levels of interaction. The resources not only support the learning process, but also offer you the opportunity to practise new knowledge.

Tip: The more exercises you do, the greater the chance that you will achieve success in mathematics.

Structure of themes Learning is a complex process. Millions of brain cells and neural pathways in our brains work together to store new information in the long-term memory so that we will be able to remember it later on. Long-term memory is not our only type of memory and when we learn, our working memory is just as important. Working memory is different from long-term memory and has a limited capacity. This means that one’s working memory can only handle a small amount of new information at a time.

When one learns mathematics, there is a lot of new information your brain needs to process, which can easily exhaust your working memory. The study guides have been written and compiled in such a way that it does not overtax the working memory and therefore simplifies the process of learning mathematics.

2

Revision

Exercises

This may involve one of the following:

The exercises give you the opportunity to practise the concepts and procedures taught. It is important to try and complete all exercises. Complete solutions are provided in the facilitator’s guides.

1.

revision of the concepts, definitions and procedures required as previous knowledge, 2. an exercise or activity with solutions so that you can test your previous knowledge yourself, or 3. a combination of the above.

Do not neglect this revision. It is important to work through this section thoroughly. Mathematical concepts often follow on one another and if basic knowledge is lacking or has not been mastered sufficiently, this will handicap the formation of new knowledge.

Questions usually progress from easy (in order to master and practise basic concepts and procedures) to difficult (more complex operations).

Mixed exercises are also provided, where you get the opportunity to practise different concepts and procedures and integrate these with previous themes.

Summary of theme

Following the introductory part of the theme, new knowledge is dealt with in sub-themes. Each sub-theme has the following structure:

Here you will find a summary of what you should have mastered in the theme.

This is expressed in more formal mathematical language in order to be in keeping with the CAPS (the curriculum statement).

SUB-THEME

Introduction

End of theme exercise

New concepts and procedures are explained. Relevant previous knowledge is also dealt with here if necessary.

This is a mixed exercise involving all concepts and procedures dealt with in the theme, where this work can also be integrated with previous work. The degree of difficulty of this exercise varies. It is important that you try and complete all the exercises. Complete solutions can be found in the facilitator’s guide.

Worked examples

Mixed exercises such as these in this textbook form a very important component of mastering mathematics. There is a big difference between the ability to recognise your work and the ability to recall it. When you are able to recognise your work, you will often say “Oh, of course!” but you struggle to remember this when you are writing an examination. When you are able to recall your work, this means that you have captured that knowledge in your long-term

Worked examples show you how the new concepts and procedures are applied and help you understand and apply the newly taught concepts and procedures.

© Optimi

3

G10 – Mathematics

G10 – Mathematics – Study Guide 1/2

memory and are able to remember and use it. Mixed exercises enable you to not only recognise the work, but also recall it from your long-term memory.

Self-evaluation

When you practise the same type of sum or problem over and over, you often get lazy and do not reflect upon the exercise anymore. You are convinced that you know exactly what type of sum or problem you need to solve. But in a test or exam, all these problems are mixed up and then it might be difficult to know what to do. When mixed exercises form part of your learning process, you learn to identify and complete a sum or problem correctly. This means that you are truly prepared for tests or exams, because you can recall your work instead of merely recognising it.

In each theme, and usually following each sub-theme, there is an activity where you need to reflect critically about the extent to which you have mastered certain concepts and procedures. This activity has the following format:

Use the following scale to determine how comfortable you are with each topic in the table below:

1. Help! I don’t feel comfortable with the topic at all. I need help.

2. Alarm! I don’t feel comfortable, but I just need more time to work through the topic again.

3. OK! I feel moderately comfortable with the topic, but I still struggle sometimes.

4. Sharp! I feel comfortable with the topic.

5. Whoo-hoo, it’s party time! I feel totally comfortable with the topic and can even answer more complicated questions about it.

Complete the table.

Topic

1

2

3

4

5

Tip: Complete each self-evaluation as honestly as possible. If there are aspects which you have not mastered, revisit these and make sure that you do master them. Ask the facilitator for help. It is important not to move on to a next theme or sub-theme before you have mastered the topic involved, even if this means that you spend more time on a specific theme than recommended by the CAPS.

4

Exam papers

Supplementary books

Visit Impaq’s online platform for the assessment plan and comprehensive information about the compilation and mark allocation of tests, assignments and examinations.

Any other books can be used along with this textbook for extra exercises and explanations, including:

• Maths 4 Africa, available at www.maths4africa.co.za • The Siyavula textbook, available online for free at www.siyavula.com • Pythagoras, available at www.fisichem.co.za.

Tip: Use supplementary resources for further explanations, examples, and especially extra exercises.

Tip: Make sure that you know which themes are covered in which paper. The themes covered in the examination papers are subject to change. Always refer to the portfolio book and assessment plan for updated information about the composition of the examination papers.

Paper 1

Paper 2

Calculator

Algebraic expressions, equations and Euclidean geometry and inequalities, exponents (Theme 1, 2 measurement (Theme 8, 13 and 14) and 4) Analytical geometry (Theme 9) Number patterns (Theme 3) Trigonometry (Theme 5 and 7) Functions and graphs (Theme 6) Statistics (Theme 11) Finance and growth (Theme 10)

We recommend the CASIO fx-82ES (Plus) or CASIO fx-82ZA. However, any scientific, non-programmable and non-graphing calculator is suitable.

Probability (Theme 15)

Note:

• No graphing or programmable calculators are allowed (for example to factorise or find the roots of equations). Calculators should only be used to do standard numeric calculations and to verify calculations done by hand.

• Formula sheets are not provided during tests and final exams in Grade 10.

© Optimi

5

G10 – Mathematics

THEME 1

Different types of numbers ​ℕ = ​{1 ; 2 ; 3 ; 4 ; 5 ; …}​ =​natural numbers

NUMBERS AND ALGEBRAIC EXPRESSIONS

​ℕ​ 0​ = ​{0 ; 1 ; 2 ; 3 ; 4 ; 5 ; …}​ =​whole numbers ​ℤ = ​{… ; − 2 ; − 1 ; 0 ; 1 ; 2 ; 3 ; …}​ =​ integers

Introduction

_____________ ​ℚ = ​{ numbers that can be written as ​

​ ​= rational numbers​ a non-zero integer} an integer

​ℚ′ ​ = ​{ numbers that cannot be written as a ​ _____________

​}​ = irrational numbers​ non-zero integer

In this theme you will learn more about: • • • • • •

an integer

different types of numbers how to estimate the values of certain numbers how to round off numbers how to multiply algebraic expressions how to find factors of algebraic expressions how to simplify algebraic fractions.

(non-terminating and non-repetitive decimal numbers)

​ℝ = ​{rational and irrational numbers}​ =​real numbers

​ℝ′ ​ = ​{ numbers that do not exist in the real number system}​ =​ non-real numbers

Prior knowledge

Summary of the real number system

In order to master this theme, you should already know: • • • •

Real numbers

what types of numbers there are how we classify numbers how to multiply simple algebraic expressions how to simplify simple fractions.

Rational numbers

Integers

1.1 THE NUMBER SYSTEM

Fractures

• Negative integers • Zero • Positive integers (natural numbers)

Introduction This sub-theme is a summary of work covered in Grades 8 and 9. If you should find it difficult to complete this section, you should first revise the work covered in these grades.

© Optimi

Irrational numbers

REMEMBER

_0 𝑥​ ​ = 0

7

_

​√ − 3 ​ is non-real

_𝑥 ​ is undefined​ 0 G10 – Mathematics

Theme 1: Numbers and algebraic expressions

G10 – Mathematics – Study Guide 1/2 Note

Worked example 2

• A rational number is any number that can be written as ​_ b a ​, where ​a​and ​b​ are integers.

• The following are rational numbers: ◦ fractions of which both the numerator and denominator are integers, e.g. 73​_ ​ ◦ integers, e.g. − ​ 5​ ◦ decimal numbers that end, e.g. ​0,125​

Rewrite ​2,5​1˙ ​2˙ ​ as a common fraction.

Solution

Let ​y = 2,512121212…​

◦ decimal numbers that repeat, e.g. ​0, 151515…​

• Irrational numbers are not rational. They cannot be written with an integer numerator and denominator, e.g. ​0,8672345…​ • If the ​nth ​ root of a number cannot be written as a rational value, this ​nth ​ 3 _ root is called a surd, e.g. ​√ 5 ​​.

​∴ 1 000y = 2512,121212…​ ​ − 10y = 25,121212…​ ​

​∴

Rewrite ​0, ​1˙ ​2˙ ​ as a common fraction.

Solution

In order to rewrite a recurring fraction as a common fraction, you need to manipulate the recurring fraction to lose the recurring “tail”.

Let ​𝑥 = 0,1212121212…​

​∴ 100𝑥 = 12,1212121212…​

​× 100​to get integer ​+​recurring “tail”

​ −𝑥 = 0,1212121212…​

Subtract

_4 ​∴ 𝑥 = 33 ​​

Simplify

​ 99𝑥 = 12​ ​∴

𝑥 = ​_ 12 ​​ 99

8

Remember that the 5 ​ ​is not recurring

990y = 2487,000000…​

2 487 ​∴ y = ​_ 990 ​​

Worked example 1

​× 1 000​and × ​ 10​to get integer ​+​recurring “tail”

829 y = ​_ 330 ​​

Always remember to simplify completely

_ ​ 02 ​

Worked example 3 Using your knowledge of the number system, complete the following table by making a  in the appropriate block(s):

3

_ ​ 17 ​

_

ℚ

ℚ'

ℝ

√

_ ​ 02 ​ __ 9 ​​ _ ​16 ​

√

1

​​√ 50 ​ _

It exists (your calculator does not give you a math error message); therefore it is a real number (ℝ).

​​√ 50 ​ =3,684031499… (non-finite, non-recurring decimal fraction)

In order to determine where these numbers fit into the number system, you can use your calculator to find the decimal fraction where applicable:

_

It exists (your calculator does not give you a math error message); therefore it is a real number (ℝ). ​ ​_ = 3 ​​ This number is written in the form ​​_ ab ​therefore it is a rational number (ℚ).

_

3

It exists (your calculator does not give you a math error message); therefore it is a real number (ℝ).

0,​​3̇ ​ = 0,333333333… (non-finite, recurring decimal fraction)

0,​​3̇ ​

_ ​ 17 ​

This number is written in the form ​​_ ab ​therefore it is a rational number (ℚ).

9 3 ​​ _ ​16 ​ = ​ ​ _ ​ 4 This number is written in the form ​​_ ab ​therefore it is a rational number (ℚ).

​​ √ 9 ​

Solution

__

= 0 (zero divided by any non-zero number = zero)

This number is written in the form _ ​ ab ​; therefore it is a rational number (ℚ).

It exists (your calculator does not give you a math error message); therefore it is a real number (ℝ).

This number cannot be written in the form ​​ _ab ​therefore it is an irrational number (ℚ').

It exists (your calculator does not give you a math error message); therefore it is a real number (ℝ).

​​ √ 9 ​ = 0,2080083823… … (non-finite, non-recurring decimal fraction) This number cannot be written in the form ​​ _ab ​therefore it is an irrational number (ℚ').

It exists (your calculator does not give you a math error message); therefore it is a real number (ℝ). © Optimi

9

G10 – Mathematics

Theme 1: Numbers and algebraic expressions

G10 – Mathematics – Study Guide 1/2

ℚ

ℚ'

ℝ

ü

7.

Rewrite the following as common fractions:

ü

ü

7.2

3,1​​5̇ ​6​ ̇

_ ​ 17 ​

ü

_ ​ 02 ​ __ 9 ​​ _ ​16 ​

ü

ü

ü

ü

ü

ü

3

_

​​ √ 9 ​

√

0,​​3̇ ​

_

​​√ 50 ​

ü

7.1 8.

0,​​6​̇

For which value(s) of 𝑥 will f(𝑥) be non-real if: _

f(𝑥) = ​ ___ 11 ​ 9 ​ ​ – 𝑥 and 𝑥∈{–5; 0; 11}?

√

1.2 BETWEEN WHICH TWO INTEGERS DOES A SURD FALL?

ü

Introduction

Note that real numbers (ℝ) are either rational (ℚ) or irrational (ℚ') numbers.

If the n ​ ​th root of a number cannot be simplified to a rational number, we call it _

6

Exercise 1.1: The number system

be simplified to the rational number ​2​. n

1.

Is the number zero a positive or negative number?

3.

What type of number is ​√ −8 ​?

2.

4.

5.

6.

_

_

3

_

2

_

​ ​to be 2 ​ ​, so we usually do not write ​​√ a ​​. √ 7 ​​ or ​​√ 5 ​​. It is very common for n

_

_

Instead, we write the surd as simply ​√ a ​. This is called the square root of a ​ ​.

It is sometimes useful to know the approximate value of a surd without having to use a calculator.

_ _

What type of number is ​√ 8 ​ ?

3

_

Consider surds of the form ​√ a ​ , where a ​ ​is any positive number, for example​

What type of number is ​​√ 8 ​​?

3

_

a surd. For example, ​√ 2 ​and ​√ 3 ​ are surds, but ​√ 4 ​ is not a surd because it can

_

For example, let us estimate where ​√ 3 ​ ​lies on the number line:

_

_

Using a calculator, we know that ​√ 3 ​ = 1,73205… ​

What type of number is ​√ −8 ​?

_

It is easy to see that ​√ 3 ​ is greater than ​1​and smaller than ​2​.

Without using a calculator, determine all the number types that 2 ___ ​ 10 ​​ 27 belongs to.

10

_

But to estimate the values of other surds, such as ​​√ 18 ​​, without using a calculator, you should first understand the following: n

_

n

Exercise 1.2: Estimating surds

_

• If ​a​and ​b ​are positive integers and a ​ < b​, then ​√ a ​ < ​√ b ​​.

• A perfect square is the number obtained when an integer is squared.

Do the exercise without using a calculator.

For example, ​9​is a perfect square since ​3​ 2​ = 9​.

• A perfect cube is a number which is the cube of an integer.

Estimate ​√ 10 ​ correct to one decimal place.

3

_

Find two consecutive integers such that ​√ 49 ​ lies between them. _

Determine between which two consecutive integers ​√ 18 ​​ lies. _

1.3 ROUNDING OF REAL NUMBERS

_

Determine between which two integers the irrational number ​√ 62 ​​ lies.

Introduction Rounding numbers makes them simpler and easier to use. It is often just easier to work with rounded numbers.

Solution

Find the two perfect squares to the left of (just smaller than) and to the right of

Rounding numbers means adjusting the digits (up or down) in order to make rough calculations easier. The result will be an estimated answer rather than an exact one.

(just bigger than) ​62 ​on the number line.

• The perfect square to the left of ​62​is 4 ​ 9​.

How to round numbers:

• The perfect square to the right of ​62​is 6 ​ 4​.

• When rounding to a required number of places, the next decimal digit is considered, e.g. if you are required to round to three decimal places, consider the 4 ​ th ​ decimal digit.

Now find the square roots of these perfect squares: _

• ​√ 49 ​ = 7​

• If the next digit is less than ​5​, the previous decimal digit stays as it is.

• If the next digit is ​5​or more, the previous decimal digit is increased by one.

• If you are asked to round to three decimal places, you need to have three digits after the decimal comma (even if these are zeros).

• ​√ 64 ​ = 8​

_

Therefore, ​√ 62 ​ lies between 7 ​ ​and ​8​. _

​∴ 7 < ​√ 62 ​ < 8​

© Optimi

4.

3.

Worked example 4

_

Find two consecutive integers such that ​√ 26 ​ lies between them.

2.

For example, ​27​is a perfect cube, because ​3​ 3​ = 27​.

_

1.

11

G10 – Mathematics

Theme 1: Numbers and algebraic expressions

G10 – Mathematics – Study Guide 1/2

Worked example 5

Self-evaluation Use the following scale to determine how comfortable you are with each topic in the table that follows:

Round ​2,6003​to three decimal places.

1. Help! I am not at all comfortable with the topic; I need help.

2. Alarm! I am not comfortable with the topic; however, I merely need more time to go through the topic again.

3. OK! I am fairly comfortable with the topic, but do get stuck occasionally.

4. Sharp! I am comfortable with the topic.

5. Whoohoo, it's party time! I am completely comfortable with the topic, and can even answer more difficult questions on this.

Solution

Look at the 4 decimal digit: 3<5 3rd digit will stay the same. th

​2,6003​

​= 2,600​

Worked example 6

Complete the table.

Round ​473,78​to the nearest integer.

Topic

​= 474​

Look at the 1st decimal digit: 7>5 Units digit will become 1 more.

I can write any recurring decimal fraction as an ordinary fraction.

I can determine between which two integers a given surd falls.

Exercise 1.3: Rounding numbers

I can round real numbers to a certain degree of accuracy.

Round the following numbers to the indicated degree of accuracy: 1.

​12,507​to two decimal places

4.

​28,995​to two decimal places

2.

3.

5.

​36,8121212​to the nearest integer ​− 48,2291​to three decimal places ​− 185,02​to one decimal place

2

3

4

5

I can classify any real number as rational or irrational.

Solution ​473,78​

1

12

Consider the algebraic expression ​3a​​ 4​ + 2 ​a​ 2​ − 3a + 8​.

1.4 MULTIPLICATION OF ALGEBRAIC EXPRESSIONS

Terminology

Introduction

Coefficient

In this sub-theme we deal with multiplication of algebraic expressions.

1.4.1 Algebraic expressions

An algebraic expression contains letter symbols and numbers, e.g. ​2 ​𝑥​ ​ − 3𝑥y + ​y​ ​ − 5​.

2

3

Constant

Note the difference between an algebraic expression and an algebraic equation:

Degree of expression

• In an algebraic equation, two expressions are equated (using an equal sign).

Examples: ◦ ​2x − 5 = 0​ ◦ ​3 ​x​ 2​ − 5x = 7x + 2​

Number of terms Variable

• An algebraic expression does not have an equal sign. Examples: ◦ ​2x − 5​ ◦ ​3 ​x​ 2​ − 5x​

Description

Example

The number (including its sign) multiplied by the variable

Coefficient of ​a ​ ​ 4​​is + ​ 3​

The term without any variable (including its sign)

Constant is ​+ 8​

The highest exponential value

Terms are separated by plus (​+​) and minus (​−​) signs

The letter symbols that appear in the algebraic expression. Note: Variables can have any permissible value.

Coefficient of ​a ​ ​ 2​​is + ​ 2​ Coefficient of a ​ ​is − ​ 3​

Degree of expression is 4 ​​ Number of terms is 4 ​​ Variable is ​a​

Algebraic expressions are named according to the number of terms in the expression:

• Algebraic equations can be solved, i.e. we can find the unique value(s) of the unknown (say ​x​) for which the equation is true.

• Algebraic expressions cannot be solved. In an algebraic expression, the variable(s) (say ​x​) can take on any value that is allowed.

Monomial Binomial Trinomial Polynomial

You should know the following terminology regarding algebraic expressions:

– – – –

1.4.2 Products

1 term 2 terms 3 terms two or more terms

Multiplying a monomial by a polynomial: • Always use the distributive property to remove the brackets.

• Use exponent laws where required. (Refer to exponent laws in Theme 2 if you are unsure of these.)

• Simplify like terms. © Optimi

13

G10 – Mathematics

Theme 1: Numbers and algebraic expressions

G10 – Mathematics – Study Guide 1/2

Multiplying a binomial by a binomial:

Multiply the number outside the bracket by each term inside the bracket:

Use FOIL to determine the product:

2(𝑥 – 4)

F O I L

Worked example 7

1. 𝑥(𝑥 +2) = (𝑥 × 𝑥)+(𝑥 × 2) 2. – 3𝑦² (𝑥² – 2) = (–3𝑦² × 𝑥²) + (–3𝑦² × –2)

Distributive law

3. 2b(3c + d) – 3(b – c + d)

Distributive law

= – 3𝑥² 𝑦² + 6𝑦²

multiply the first terms of the binomials multiply 1st term of first binomial by last term of second binomial multiply last term of first binomial by 1st term of second binomial multiply the last terms of the binomials

Worked example 8

Using distributive law

= 𝑥² + 2𝑥

First terms Outer terms Inner terms Last terms

= 2b × 3c + 2b × d – 3 × b – 3 × (– c) – 3 × d = 6bc + 2bd – 3b + 3c – 3d

1.

​(​𝑥 − 2​)​(​𝑥 + 4​)​​

F: (𝑥) × (𝑥) = 𝑥² O: (𝑥) × ( + 4) = 4𝑥 I: ( – 2) × (𝑥) = –2𝑥 L: ( – 2) × ( + 4) = –8

2.

​(3a − 4b)​(​2a + 3b​)​

F: 3𝑎×2𝑎=6𝑎² O: 3𝑎×3b=9ab I: -4b×2𝑎=-8𝑎b L: -4b×3b=-12b2

​= 𝑥​​ 2​ + 4𝑥 − 2𝑥 − 8​ ​= ​𝑥​ 2​ + 2𝑥 − 8​

​= ​6a​​ 2​ + 9ab − 8ab − 12 ​b​ 2​​

​= 6 ​a​ ​ + ab − 12 ​b​ ​​ 2

2

​​ ​ ​ ​

​​

Use “shortcut” method – only applicable if you have a binomial square: i) Square the first term. ii) Multiply the first term and second term, and multiply the answer by​ 2​. iii) Square the second term.

REMEMBER

(+) × (+) = (+) (–) × (–) = (+) (+) × (–) = (–) (–) × (+) = (–)

Why do you think this shortcut method works? (Hint: Use FOIL method.)

14

Worked example 9

Exercise 1.4: Multiplication of algebraic expressions

1. (2𝑥 – 3𝑦)²

= 4𝑥² – 12𝑥𝑦 + 9𝑦²

(2𝑥 × 2𝑥) = 4𝑥2 F

2 ×(2𝑥 × -3y) = -12𝑥y O+I

2. (2𝑥 + 𝑦)²

= 4𝑥² + 4𝑥𝑦 + 𝑦²

1.

1.1 1.2

(-3y × -3y) = 9y2 L

1.3

(i) 2𝑥 × 2𝑥 = 4𝑥² (ii) 2𝑥 × y × 2 = 4𝑥y (iii) y × y = y²

1.4 1.5

Multiplying polynomials: i)

ii) iii)

1.6

Work from left to right and multiply every term of first polynomial by every term of second polynomial.

Use exponent laws where necessary.

Simplify like terms.

1.7 1.8

Worked example 10

​(𝑥 + y)​​(​2𝑥 − 3y + 8​)​​

​= ​2𝑥​​ 2​ − 3𝑥y + 8𝑥 + 2𝑥y − 3 ​y​ 2​ + 8y​ ​= 2 ​𝑥​ 2​ − 𝑥y + 8𝑥 + 8y − ​3y​​ 2​

1.9

Simplify the following: 2​​(𝑥 − 1)​​− 3​​(𝑥 + 4)​​ ​−​(𝑥​𝑦​​ 2​)​(−3𝑥)​

​(2𝑥 − 5)​​(2𝑥 + 5)​​

​

​​(2𝑥 − 7)​2​​

​​(𝑥 ​ 2​​ ​ – ​𝑦​​​2​​)2​​​​​ ​ ​​(3 − 𝑥)​2​​

​​(−𝑦 − 3)​2​​

−7 ​​(7𝑥 − 1)​​​2​​ ​

​(21​_ ​𝑥 − 4)​

​(8𝑥 − 10)​​

1.10 −​​(𝑥 − 3)​​(𝑥 + 8)​​− ​​(𝑥 + 2)​​ 2​​

Multiply 1st term of binomial (i) 𝑥 × 2𝑥 = 2𝑥² (ii) 𝑥 × – 3y = – 3𝑥y (iii) 𝑥 × 8 = 8𝑥 Multiply 2nd term of binomial (iv) y ×2𝑥 = 2𝑥y (v) y ×-3y = 3y2 (vi) y × 8 = 8y

1.11 2​​(𝑥 + 1)​​(𝑥 − 1)​​− ​​(𝑥 − 2)​​ 2​​ 1.12 4 − 2 ​(𝑥 + 3)​​

1.13 ​​{2𝑥 ​(𝑥 − 1)​+ 2𝑥}​​​{−𝑥 ​(𝑥 − 1)​+ 𝑥}​​ 1.14 ​(​𝑥 − 2​)​ 3​

1.15 ​(𝑎 + 3)​​(𝑎 − 3)​​ © Optimi

15

G10 – Mathematics

Theme 1: Numbers and algebraic expressions

G10 – Mathematics – Study Guide 1/2 1.16 ​(2𝑎 − 5)​​(2𝑎 + 5)​​

2.12

​(𝑎 −b)​​​ 2​​​​(𝑎 + b)​​​ 2​

1.18 ​(0,1 − 3b)​​(0,1 + 3b)​​

2.14

​(𝑎 + b + c)​​(𝑎 + b − c)​​

1.17 ​(−𝑎 + 6)​​(−𝑎 − 6)​​

2.13 2.15

1.19 ​(30𝑎bc − 1)​​(30𝑎bc + 1)​

2.

2.1 2.2 2.3 2.4

​​(−m + 1)​​ 2​

3.2 3.3

​​(1​​_2 ​​𝑥 − 2𝑦)​​ 2​

2.7

​(−𝑦 − 3)​​​ 2​​​ ​​(−3)​​

2.9

3.1

​​(2g − 4m)​​ 2​ − ​​(2𝑥 − 7)​​ 2​

2.8

3.

​​(−𝑥 − 2)​​ 2​

2.5 2.6

2.16

Simplify the following using the short method:

3.4

−2 ​(−𝑥 + 2)​​​ 2​​​

3.5

​(𝑥 − 4)​​​(−4)​(𝑥 − 4)​​​

​(𝑥 ​ ​ ​ + 16)​​(𝑥 + 4)​​(𝑥 − 4)​​

2.10 ​(𝑥 − 2)​​(𝑥 + 2)​​​(𝑥 + 3)​​ ​ 2.11

2

​𝑥​​ 2​(𝑥 ​ ​​ 2​​ 𝑦​​ 2​ + ​𝑥​​ 2​)​(​ 𝑥𝑦 −𝑥 )​​(𝑥𝑦 + 𝑥)​​

2𝑎b​​(𝑎b − ​𝑎2​b)​​− ​𝑎2​​ ​b2​− ​(𝑎b − b)​​(𝑎b + b)​​

Simplify the following using the short method: ​​​(_ ​ 3𝑥 ​− ​_ 3𝑥 ) ​ ​ ​​​​​(_​ ​_ ​)​ ​​​​​(__​ ​ ​ ​​ )​ ________ ​ ( ) ​ _​​ _ _​ _​​ _ _ ​​ _​ ​

1 1 ​​(𝑎− ​ _ ​ ​​ 𝑎+ ​ _ ​ ​​ 4) 2 )(

​​(​ ​ ​)​ ​

​​ ​ ​ ​ ​ ​

​​(2​_ 1 ​b − 3e)​​(1​_ 1 ​ ​​ 4) 2 ​​(_ ​ 1 ​a − ​_ 1 ​ (​​ 2a − 3)​​ 2 3)

​​(_ ​ a ​− 2)​​(__ ​ ​a4​ ​ ​+ a + 4)​​ 2 2

3.6

1 ​​​(_ ​ 1 ​𝑥 − ​ _ ​ 𝑦 ​​ ​​ 2 3 )

3.8

m _ ​​(_ ​ 1 ​− m)​​(_ ​ 1 ​+ m)​− ​ ___ ​ ​ 1 ​ + ​mn​​ 2​)​​ 2 (m n n

3.7

2

​(𝑎 ​ ​ 2​​b​​ 2 + ​𝑎​​ 2​)​(​ 𝑎b − 𝑎)​​(𝑎b + 𝑎)​​

2

​​​(_ ​ 1 ​+ ​_ 1 ​) ​ ​− 2​​(_ ​ 1 ​) ​​(_ ​ 1 ​)​− ​(_ ​ 1 ​− ​_ 1 ​) ​​​​(_ ​ 1 ​+ ​_ 1 ​) ​​ n m m n n m n m 2

3.9 Using a calculator:

​​(𝑥 − 1)​​​2 ​​(𝑥 + 1)​​​2​

​n​ ​

​​(2,3 − 1,7m)​​(3,5m − 0,07)​​

3.10 ​​​(0,2𝑥 − 0,3𝑦)​ 2​​ 16

Without a calculator:

​​(_ ​ 23 ​ – ​_ 17m ​ ​​​​ _ ​ 35m ​ – ​_ 7 ​ ​​ 10 10 )( 10 100)

1.5 FACTORISING ALGEBRAIC EXPRESSIONS

Self-evaluation

Introduction

Use the following scale to determine how comfortable you are with each topic in the table that follows: 1. Help! I am not at all comfortable with the topic; I need help.

2. Alarm! I am not comfortable with the topic; however, I merely need more time to go through the topic again.

3. OK! I am fairly comfortable with the topic, but do get stuck occasionally.

4. Sharp! I am comfortable with the topic.

5. Whoohoo, it's party time! I am completely comfortable with the topic, and can even answer more difficult questions on this.

Complete the table.

Topic

1

2

3

4

It is important to understand the difference between factors and products.

What is a factor?

• The factors of a number are what we can multiply to get the number.

• We can also think of a factor of a number as something that we can divide into that number without leaving a remainder.

What is a product?

• This is the answer when two or more factors are multiplied.

Examples:

5

24 = 4 × 6

I know the difference between an algebraic expression and an algebraic equation.

I know the following terminology regarding algebraic expressions, and can apply it: • Coefficient • Constant • Variable • Degree of an expression • Number of terms I can multiply algebraic expressions.

© Optimi

9=3×3

4 and 6 are the factors of 24.

[4 and 6 can be divided into 24 without leaving a remainder.]

24 is the product of 4 and 6.

3 is a factor of 9.

[3 can be divided into 9 without leaving a remainder.]

Multiplication and factorisation are opposite (inverse) procedures.

If we have the product, we can find the factors that have been multiplied to give the product.

Factorise = find the factors

9 is the product of 3 and 3 (32).

Types of factorisation: • • • • • 17

Common factor Difference of squares Quadratic trinomial Grouping in pairs Sum and difference of cubes G10 – Mathematics

Theme 1: Numbers and algebraic expressions

G10 – Mathematics – Study Guide 1/2 Let us first revise Grade 9 factorisation, namely taking out a common factor and the difference of squares. We shall also revise quadratic trinomials later on.

Worked example 11

1.5.1 Common factor What is a common factor? A common factor is a number that can be divided into two or more different numbers without leaving a remainder. Numbers often share more than one common factor.

Factorise:

–3𝑥³ + 6𝑥𝑦 – 12𝑥²

= –3.𝑥.𝑥.𝑥 + 2.3.𝑥.𝑦 – 3.4.𝑥.𝑥 = –3.𝑥(𝑥.𝑥 – 2.𝑦 + 4.𝑥)

For example, ​2​is a common factor of ​4​and ​12​.

You should always check your answer by multiplying the factors to see if they get the original expression, i.e. the given product.

In this case, ​4​is the highest common factor (HCF). ​𝑥​ is a common factor of ​𝑥​ 2​and ​𝑥​ 3​.

This should become a habit.

However, ​𝑥​ 2​is also a common factor of ​𝑥​ 2​and ​𝑥​ 3​. In this case, ​𝑥​ 2​is the highest common factor.

–3𝑥(𝑥² – 2𝑦 + 4𝑥) = –3𝑥³ + 6𝑥𝑦 – 12𝑥²

Factorise using a common factor of an algebraic expression such as ​4 ​𝑥​ ​ + 12 ​𝑥​ ​​:

iii)

Take out the negative sign if the first term is negative

= –3𝑥(𝑥² – 2𝑦 + 4𝑥)

However, ​4​is also a common factor of ​4​and ​12​.

i) ii)

Determine the highest common factor (3𝑥)

2

3

Determine the highest common factor of the terms (​4 ​𝑥​ 2​in this case).

The second factor is obtained by dividing the common factor into each term of the original algebraic expression. This gives 1 ​ + 3𝑥​. Therefore,​ 4 ​𝑥​​ 2​ + 12 ​𝑥​​ 3​is factorised as ​4 ​𝑥​​ 2​​(​1 + 3𝑥​)​​​​. Note that the second factor is written in brackets.

So, the factors of 4 ​ ​𝑥​​ 2​ + 12 ​𝑥​​ 3​are ​4 ​𝑥​​ 2​and ​1 + 3𝑥​. Therefore, if we multiply ​4 ​𝑥​​ 2​and ​1 + 3𝑥​, we will obtain the original expression, ​4 ​𝑥​​ 2​ + 12 ​𝑥​​ 3​​.

Original expression given

Worked example 12

Factorise:

​3𝑥(𝑥 – 1) – 2(𝑥 – 1) = (𝑥 – 1)(3𝑥 – 2)

​Check your answer.

18

In this case, the common factor is a binomial

Conversely, to find the factors of a difference of squares, we follow this procedure:

Worked example 13 Factorise:

3𝑥(𝑥 – 1) – 2(1 – 𝑥)

In this case it might seem as if the common factor is (𝑥 – 1), but the factors (𝑥 – 1) and (1 – 𝑥) are not exactly the same.

However, we can write (1 – 𝑥) as –(𝑥 – 1). [Check this by removing the brackets.]

REMEMBER

Add ​+​in one bracket and ​−​in the other.

_

_

_

Worked example 14

= (4𝑥 + 5𝑦)(4𝑥 – 5𝑦)

_ _ _ _ ​​​(​​ ​√ 16 ​𝑥​​ 2 ​ ​​ ​​ ​√ ​25y​​ 2 ​ )​​​(​​ ​√ ​16𝑥​​ 2 ​ ​​ ​​ ​√ ​ 25y​​ 2 ​ )​​​​

Worked example 15

3𝑥(𝑥 – 1) – 2(1 – 𝑥)

= 3𝑥(𝑥 – 1) + 2(𝑥 – 1) – 2(1 – 𝑥) = 2(𝑥 – 1) = (𝑥 – 1)(3𝑥 + 2)

Factorise completely:

Check your answer by multiplying the factors

8𝑎² – 32b²

1.5.2 Difference of two squares A difference of two squares is a squared number/expression subtracted from another squared number/expression.

Examples: ​5​ 2​ − ​4​ 2​or ​𝑥​ 2​ − ​y​ 2​

= 2(4𝑎² – 16b²)

= 2(2𝑎 + 4b)(2𝑎 – 4b)

"Difference" means "minus".

​(𝑥 − y)​(𝑥 + y)​ = ​𝑥​ ​ + 𝑥y − 𝑥y − ​y​ ​​ 2

(8 = 2 × 4) and (32 = 2 × 16) Difference of squares

Take out common factor first Factorise difference of squares

REMEMBER

Let us see what happens when we find the product of ​(𝑥 − y)​and ​(​𝑥 + y​)​​:

© Optimi

ii)

_

​(​ ​√ ​1​ st​term ​ ​ ​ ​√ ​2​ nd​term ​) ​(​ ​√ ​1​ st​term ​ ​ ​ ​√ ​2​ nd​term ​) ​

​​16𝑥​​ 2​ − 25 ​y​ 2​​

Therefore

​= ​𝑥​ 2​ − ​y​ 2​​

Create two brackets:

Factorise:

Since variables such as x represent numbers, the same principles that apply to numbers also apply to algebraic expressions and variables. Example: 5 = (7 – 2) = – (2 – 7) = – ( – 5) = 5

2

i)

Always factorise until it is impossible to factorise any further.

Multiply a binomial by a binomial

Product is the difference of two squares

19

G10 – Mathematics

Theme 1: Numbers and algebraic expressions

G10 – Mathematics – Study Guide 1/2

Worked example 16 Factorise completely: (𝑥 – 1)² – 4

= [(𝑥 – 1) + 2][(𝑥 – 1) – 2] = (𝑥 – 1 + 2)(𝑥 – 1 – 2) = (𝑥 + 1)(𝑥 – 3)

Worked example 17

First term is a binomial _ ​​​√(​​ ​​𝑥 − 1​)​​​​ 2 ​ ​ = ​(​​𝑥 − 1​)​​​​

Solution

_

and √ ​​ 4 ​ = 2​

​a − b − a𝑥 + b𝑥​

Simplify

Group together ​(​a − b​)​​ and (​ ​ − a𝑥 + b𝑥)​ ​

​= ​(a − b)​ − 𝑥​(​a − b​)​​

Take out − ​ 𝑥​as a common factor from − ​ a𝑥 + b𝑥​

_ Remember that a − b ​a − b​ = 1​

​

​= ​(​a − b​)​(​1 − 𝑥​)​​

1.5.3 Grouping terms (new work) When you are doing basic factorisation of algebraic expressions, you usually start by taking out a common factor from every term in the expression.

What happens if the given expression has four or five terms, but there is no common factor?

Worked example 18

When you have four or five terms with no common factor for all of them, consider factorising by grouping the terms.

Check your answer by e𝑥panding the brackets

Factorise: ​a​ 2​ + a​(3 + b)​ + 3b​

To factorise by grouping terms: i) ii)

Note: –a𝑥 + b𝑥 = –(a𝑥 – b𝑥) = –𝑥(a – b)

Factorise: a ​ − b − a𝑥 + b𝑥​

Solution

First split the terms in the expression into two groups.

Then factorise each group of terms separately. If set up correctly, you should get a common factor in binomial form.

𝑎² + 𝑎(3 + b) + 3b

= 𝑎² + 3𝑎 + 𝑎b + 3b = (𝑎² + 𝑎b) + (3𝑎 + 3b) = 𝑎(𝑎 + b) + 3(𝑎 + b)

= (𝑎 + b)(𝑎 + 3)

20

Remove brackets using distributive law to get four terms Group (𝑎² + 𝑎b) and (3𝑎 + 3b) together

Take out a ​ ​as a common factor from the ​1st​ pair; take out 3 ​ ​as a common factor from the 2 ​ nd ​ pair. ​(​a + b​)​is the common factor in each binomial Take out (​ ​a + b​)​as the common factor