Here are the step-by-step workings:
a) 3 + 1 = 4
8 6
b) (2x - 2) + (x - 1)
(x2 + x)(x2 - 1)
To add the fractions, we need a common denominator.
The lowest common denominator of 8, 6, x2 + x, and x2 - 1 is 24(x2 + x)(x2 - 1).
a) (3×24)(x2 + x)(x2 - 1) + (1×24)(x2 + x)(x2 - 1)
24(x2 + x)(x2 - 1) 24(x2 + x
This document covers several topics in calculus including differentiation, implicit differentiation, transformations, trigonometry, vectors, and integration.
This document discusses properties of inverse trigonometric functions including derivatives of inverse trigonometric functions and general integration formulas that can be applied to them. Examples are provided to illustrate the use of derivatives to find the inverse trigonometric functions and how integration formulas can be generalized for inverse trigonometric functions.
The document provides revision notes on various mathematics topics including:
1) Binomial expansions, partial fractions, trigonometry formulas, and techniques for integration like volumes of revolution.
2) Parametric equations, vectors, vector equations, planes, and differential equations.
3) Details are given for solving problems involving these topics, such as using compound angle formulas, rewriting algebraic fractions as partial fractions, and separating variables to solve first-order differential equations.
This document provides instructions to expand binomial expressions up to specific terms. It asks the reader to:
1) Expand (1+x)1/2 up to x3 and then use that to expand (1-x)1/2, stating the interval of validity.
2) Find the first four terms of (1+x)1/3 and then expand (1-x)1/3 to x10, stating the interval of validity.
This document provides information about a book that is designed to help students achieve their best possible grade in Edexcel GCE Mathematics Core 1 and Core 2 units. It separates the material into the two units and includes features like key points, worked examples, exercises and revision questions to aid learning. At the end, it provides full solutions, a glossary and list of formulae to support student understanding and exam preparation.
CS 177 – Project #1 Summer 2015 Due Date =========.docxfaithxdunce63732
CS 177 – Project #1
Summer 2015
Due Date:
==========
This project is due Thursday July 9th before 11:59pm.
This assignment is an individual project and should be completed on your own using only your own
personally written code. You will submit one (1) copy of the completed Python program to the
Project 1 assignment on Blackboard. The completed file will include your name, the name of the
project and a description of its functionality and purpose of in the comments header. The file should
be named ”Project-1.py”.
This project will be the foundation of future assignments this semester, so it is important that you
maximize your program’s functionality.
Problem Description: Simulating the Movements of Cells in a Microscope
==============================================================
In 2014 Virginia scientist Eric Betzig won a Nobel Prize for his research in microscope technology.
Since receiving the award, Betzig has improved the technology so that cell functions, growth and
even movements can now be seen in real time while minimizing the damage caused by prior
methods. This allows the direct study of living nerve cells forming synapses in the brain, cells
undergoing mitosis and internal cell functions like protein translation and mitochondrial movements.
Your assignment is to write a Python program that graphically simulates viewing cellular organisms,
as they might be observed using Betzig’s technology. These simulated cells will be shown in a
graphics window (representing the field of view through Betzig’s microscope) and must be
animated, exhibiting behaviors based on the “Project Specifications” below. The simulation will
terminate based on user input (a mouse click) and will include two (2) types of cells, Crete and
Laelaps, (pronounced KREET and LEE-laps).
Crete cells should be represented in this simulation as three (3) small green circles with a radius of
8 pixels. These cells move nonlinearly in steps of 1-4 graphics window pixels. This makes their
movement appear jerky and random. Crete cells cannot move outside the microscope slide, (the
‘field’), so they may bump along the borders or even wander out into the middle of the field at times.
These cells have the ability to pass “through” each other.
A single red circle with a radius of 16 pixels will represent a Laelaps cell in this simulation. Laelaps
cells move across the field straight lines, appearing to ‘bounce’ off the field boundaries. Laelaps
sometimes appear to pass through other cells, however this is an optical illusion as they are very
thin and tend to slide over or under the other cells in the field of view.
Project Specifications:
====================
Graphics Window
• 500 x 500 pixel window
• White background
• 0,0 (x,y) coordinate should be set to the lower left-hand corner
Crete Cells
• Three (3) green filled circles with radius of 8 pixels
• Move in random increments between -4 and 4 pixels p.
This document provides examples and exercises about writing expressions using exponents to represent patterns of doubling, such as folding a paper in half multiple times. It includes:
1) An example of folding a paper in half multiple times and recording the number of layers as powers of 2.
2) Exercises asking students to predict folding outcomes, write expressions to represent folding a number of times, and relate the number of folds to the exponent.
3) Another example of bacterial reproduction through binary fission where the number of bacteria doubles each generation.
4) An example of calculating the volume of a rectangular solid where the dimensions are multiples of the width.
5) Additional practice problems involving doubling patterns with rice on
This project involves creating a simulation of gas atoms in a jar using MicroWorlds EX. Students draw a jar shape and create turtle atoms with random movement rules. When atoms collide with walls or each other, they bounce and slightly increase in speed. The goal is to escape through the open mouth of the jar. Students can experiment with variable like atom number, jar size, and speed increases to observe emergent behaviors. The project provides an introduction to using simulations and models to explore scientific concepts.
This document covers several topics in calculus including differentiation, implicit differentiation, transformations, trigonometry, vectors, and integration.
This document discusses properties of inverse trigonometric functions including derivatives of inverse trigonometric functions and general integration formulas that can be applied to them. Examples are provided to illustrate the use of derivatives to find the inverse trigonometric functions and how integration formulas can be generalized for inverse trigonometric functions.
The document provides revision notes on various mathematics topics including:
1) Binomial expansions, partial fractions, trigonometry formulas, and techniques for integration like volumes of revolution.
2) Parametric equations, vectors, vector equations, planes, and differential equations.
3) Details are given for solving problems involving these topics, such as using compound angle formulas, rewriting algebraic fractions as partial fractions, and separating variables to solve first-order differential equations.
This document provides instructions to expand binomial expressions up to specific terms. It asks the reader to:
1) Expand (1+x)1/2 up to x3 and then use that to expand (1-x)1/2, stating the interval of validity.
2) Find the first four terms of (1+x)1/3 and then expand (1-x)1/3 to x10, stating the interval of validity.
This document provides information about a book that is designed to help students achieve their best possible grade in Edexcel GCE Mathematics Core 1 and Core 2 units. It separates the material into the two units and includes features like key points, worked examples, exercises and revision questions to aid learning. At the end, it provides full solutions, a glossary and list of formulae to support student understanding and exam preparation.
CS 177 – Project #1 Summer 2015 Due Date =========.docxfaithxdunce63732
CS 177 – Project #1
Summer 2015
Due Date:
==========
This project is due Thursday July 9th before 11:59pm.
This assignment is an individual project and should be completed on your own using only your own
personally written code. You will submit one (1) copy of the completed Python program to the
Project 1 assignment on Blackboard. The completed file will include your name, the name of the
project and a description of its functionality and purpose of in the comments header. The file should
be named ”Project-1.py”.
This project will be the foundation of future assignments this semester, so it is important that you
maximize your program’s functionality.
Problem Description: Simulating the Movements of Cells in a Microscope
==============================================================
In 2014 Virginia scientist Eric Betzig won a Nobel Prize for his research in microscope technology.
Since receiving the award, Betzig has improved the technology so that cell functions, growth and
even movements can now be seen in real time while minimizing the damage caused by prior
methods. This allows the direct study of living nerve cells forming synapses in the brain, cells
undergoing mitosis and internal cell functions like protein translation and mitochondrial movements.
Your assignment is to write a Python program that graphically simulates viewing cellular organisms,
as they might be observed using Betzig’s technology. These simulated cells will be shown in a
graphics window (representing the field of view through Betzig’s microscope) and must be
animated, exhibiting behaviors based on the “Project Specifications” below. The simulation will
terminate based on user input (a mouse click) and will include two (2) types of cells, Crete and
Laelaps, (pronounced KREET and LEE-laps).
Crete cells should be represented in this simulation as three (3) small green circles with a radius of
8 pixels. These cells move nonlinearly in steps of 1-4 graphics window pixels. This makes their
movement appear jerky and random. Crete cells cannot move outside the microscope slide, (the
‘field’), so they may bump along the borders or even wander out into the middle of the field at times.
These cells have the ability to pass “through” each other.
A single red circle with a radius of 16 pixels will represent a Laelaps cell in this simulation. Laelaps
cells move across the field straight lines, appearing to ‘bounce’ off the field boundaries. Laelaps
sometimes appear to pass through other cells, however this is an optical illusion as they are very
thin and tend to slide over or under the other cells in the field of view.
Project Specifications:
====================
Graphics Window
• 500 x 500 pixel window
• White background
• 0,0 (x,y) coordinate should be set to the lower left-hand corner
Crete Cells
• Three (3) green filled circles with radius of 8 pixels
• Move in random increments between -4 and 4 pixels p.
This document provides examples and exercises about writing expressions using exponents to represent patterns of doubling, such as folding a paper in half multiple times. It includes:
1) An example of folding a paper in half multiple times and recording the number of layers as powers of 2.
2) Exercises asking students to predict folding outcomes, write expressions to represent folding a number of times, and relate the number of folds to the exponent.
3) Another example of bacterial reproduction through binary fission where the number of bacteria doubles each generation.
4) An example of calculating the volume of a rectangular solid where the dimensions are multiples of the width.
5) Additional practice problems involving doubling patterns with rice on
This project involves creating a simulation of gas atoms in a jar using MicroWorlds EX. Students draw a jar shape and create turtle atoms with random movement rules. When atoms collide with walls or each other, they bounce and slightly increase in speed. The goal is to escape through the open mouth of the jar. Students can experiment with variable like atom number, jar size, and speed increases to observe emergent behaviors. The project provides an introduction to using simulations and models to explore scientific concepts.
This project involves creating a simulation of gas atoms in a jar using MicroWorlds EX. Students draw a jar shape and create turtle atoms with random movement rules. When atoms collide with walls or each other, they bounce and slightly increase in speed. The goal is to escape through the open mouth of the jar. Students can experiment with variable like atom number, jar size, and speed increases to observe emergent behaviors. The project provides an introduction to using simulations and models to explore scientific concepts.
This project involves creating a simulation of gas atoms in a jar using MicroWorlds EX. Students draw a jar shape and create turtle atoms with random movement rules. When atoms collide with walls or each other, they bounce and slightly increase in speed. The goal is to escape through the open mouth of the jar. Students can experiment with jar size, number of atoms, and other variables to observe emergent behaviors. The project provides an introduction to using simulations and models to explore scientific concepts.
This document contains an overview and instructions for two modeling projects in MicroWorlds EX - Gas Atoms in a Jar and Environmental Chaos.
The Gas Atoms in a Jar project uses multiple turtles to represent atoms in a jar. The turtles move randomly and accelerate on collision with walls or other turtles. The Environmental Chaos project models predator-prey relationships with turtles representing food, prey (small fish), and predators (larger fish). Backgrounds, shapes, and rules are programmed to simulate interactions between the species. Both projects involve creating simulations through randomized turtle behavior and interactions.
This project involves creating a simulation of gas atoms in a jar using MicroWorlds EX. Students draw a jar shape and create turtle atoms with random movement rules. When atoms collide with walls or each other, they bounce and slightly increase in speed. The goal is to escape through the open mouth of the jar. Students can experiment with variable like atom number, jar size, and speed increases to observe emergent behaviors. The project provides an introduction to using simulations and models to explore scientific concepts.
This project involves creating a simulation of gas atoms in a jar using MicroWorlds EX. Students draw a jar shape and create turtle atoms with random movement rules. When atoms collide with walls or each other, they bounce and slightly increase in speed. The goal is to escape through the open mouth of the jar. Students can experiment with jar size, number of atoms, and other variables to observe emergent behaviors. The project provides an introduction to using simulations and models to explore scientific concepts.
This project involves creating a simulation of gas atoms in a jar using MicroWorlds EX. Students draw a jar shape and create turtle atoms with random movement rules. When atoms collide with walls or each other, they bounce and slightly increase in speed. The goal is to escape through the open mouth of the jar. Students can experiment with jar size, number of atoms, and other variables to observe emergent behaviors. The project provides an introduction to using simulations and models to explore scientific concepts.
This document contains an overview and instructions for two modeling projects in MicroWorlds EX - Gas Atoms in a Jar and Environmental Chaos.
In the Gas Atoms project, multiple turtles represent atoms in a jar, with rules that cause them to bounce and accelerate off walls and each other. The Environmental Chaos project models predator-prey relationships with turtles representing food, prey (small fish), and predators (larger fish). Backgrounds, shapes, and behaviors are programmed for each species. Both projects use randomness and simulate phenomena through turtle interactions.
This document summarizes an R boot camp focusing on statistics. It includes an agenda that covers introducing the lab component, R basics, descriptive statistics in R, revisiting installation instructions, and measures of variability in R. Descriptive statistics are presented as ways to characterize data through measures of central tendency, shape, and variability. Examples are provided in R for calculating the mean, median, mode, range, percentiles, variance, standard deviation, and coefficient of variation. The central limit theorem and standardizing scores are also discussed. Real-world applications of R for clean and messy data are mentioned.
This document discusses exponents and exponential expressions. It contains three examples: [1] folding a piece of paper repeatedly in half, where the number of layers doubles each time, modeled by 2n; [2] bacterial growth through binary fission, where the number of bacteria doubles each generation, also modeled by 2n; [3] the volume of a rectangular solid where the dimensions are proportional, leading to an expression of the form 6w3. Students are asked to predict patterns, model situations exponentially, write expressions and evaluate them. Exponents are shown to arise in many real-world applications involving doubling or repeated multiplication.
A work in progress - drafts to be updated and completed later. Practice with the the assessment statements from the Core component of the course that require diagrams.
This document contains information about parts of bacterial cells and calculating microscope magnification. It includes diagrams of bacterial cells and their organelles like DNA and flagella. It discusses the functions of these parts. The document also contains examples of calculating total microscope magnification by multiplying the eyepiece and objective lenses. Finally, it discusses how microscope technology has advanced over time from 1000 AD to today, allowing observation of cell structures in greater detail.
The document provides examples and steps for multiplying fractions and mixed numbers. It gives the rule for multiplying fractions by changing them to improper fractions, dividing out common factors, multiplying the numerators and denominators, and simplifying. Examples are provided for multiplying fractions, mixed numbers, and finding the area of a rectangle and triangle using fractions. Students are given practice problems to solve.
This project involves creating a simulation of gas atoms in a jar using MicroWorlds EX. Students draw a jar shape and create turtle atoms with random movement rules. When atoms collide with walls or each other, they bounce and slightly increase in speed. The goal is to escape through the open mouth of the jar. Students can experiment with variable like atom number, jar size, and speed increases to observe emergent behaviors. The project provides an introduction to using simulations and models to explore scientific concepts.
This project involves creating a simulation of gas atoms in a jar using MicroWorlds EX. Students draw a jar shape and create turtle atoms with random movement rules. When atoms collide with walls or each other, they bounce and slightly increase in speed. The goal is to escape through the open mouth of the jar. Students can experiment with jar size, number of atoms, and other variables to observe emergent behaviors. The project provides an introduction to using simulations and models to explore scientific concepts.
This document contains an overview and instructions for two modeling projects in MicroWorlds EX - Gas Atoms in a Jar and Environmental Chaos.
The Gas Atoms in a Jar project uses multiple turtles to represent atoms in a jar. The turtles move randomly and accelerate on collision with walls or other turtles. The Environmental Chaos project models predator-prey relationships with turtles representing food, prey (small fish), and predators (larger fish). Backgrounds, shapes, and rules are programmed to simulate interactions between the species. Both projects involve creating simulations through randomized turtle behavior and interactions.
This project involves creating a simulation of gas atoms in a jar using MicroWorlds EX. Students draw a jar shape and create turtle atoms with random movement rules. When atoms collide with walls or each other, they bounce and slightly increase in speed. The goal is to escape through the open mouth of the jar. Students can experiment with variable like atom number, jar size, and speed increases to observe emergent behaviors. The project provides an introduction to using simulations and models to explore scientific concepts.
This project involves creating a simulation of gas atoms in a jar using MicroWorlds EX. Students draw a jar shape and create turtle atoms with random movement rules. When atoms collide with walls or each other, they bounce and slightly increase in speed. The goal is to escape through the open mouth of the jar. Students can experiment with jar size, number of atoms, and other variables to observe emergent behaviors. The project provides an introduction to using simulations and models to explore scientific concepts.
This project involves creating a simulation of gas atoms in a jar using MicroWorlds EX. Students draw a jar shape and create turtle atoms with random movement rules. When atoms collide with walls or each other, they bounce and slightly increase in speed. The goal is to escape through the open mouth of the jar. Students can experiment with jar size, number of atoms, and other variables to observe emergent behaviors. The project provides an introduction to using simulations and models to explore scientific concepts.
This document contains an overview and instructions for two modeling projects in MicroWorlds EX - Gas Atoms in a Jar and Environmental Chaos.
In the Gas Atoms project, multiple turtles represent atoms in a jar, with rules that cause them to bounce and accelerate off walls and each other. The Environmental Chaos project models predator-prey relationships with turtles representing food, prey (small fish), and predators (larger fish). Backgrounds, shapes, and behaviors are programmed for each species. Both projects use randomness and simulate phenomena through turtle interactions.
This document summarizes an R boot camp focusing on statistics. It includes an agenda that covers introducing the lab component, R basics, descriptive statistics in R, revisiting installation instructions, and measures of variability in R. Descriptive statistics are presented as ways to characterize data through measures of central tendency, shape, and variability. Examples are provided in R for calculating the mean, median, mode, range, percentiles, variance, standard deviation, and coefficient of variation. The central limit theorem and standardizing scores are also discussed. Real-world applications of R for clean and messy data are mentioned.
This document discusses exponents and exponential expressions. It contains three examples: [1] folding a piece of paper repeatedly in half, where the number of layers doubles each time, modeled by 2n; [2] bacterial growth through binary fission, where the number of bacteria doubles each generation, also modeled by 2n; [3] the volume of a rectangular solid where the dimensions are proportional, leading to an expression of the form 6w3. Students are asked to predict patterns, model situations exponentially, write expressions and evaluate them. Exponents are shown to arise in many real-world applications involving doubling or repeated multiplication.
A work in progress - drafts to be updated and completed later. Practice with the the assessment statements from the Core component of the course that require diagrams.
This document contains information about parts of bacterial cells and calculating microscope magnification. It includes diagrams of bacterial cells and their organelles like DNA and flagella. It discusses the functions of these parts. The document also contains examples of calculating total microscope magnification by multiplying the eyepiece and objective lenses. Finally, it discusses how microscope technology has advanced over time from 1000 AD to today, allowing observation of cell structures in greater detail.
The document provides examples and steps for multiplying fractions and mixed numbers. It gives the rule for multiplying fractions by changing them to improper fractions, dividing out common factors, multiplying the numerators and denominators, and simplifying. Examples are provided for multiplying fractions, mixed numbers, and finding the area of a rectangle and triangle using fractions. Students are given practice problems to solve.
3. Contents C4
6 Partial fractions 147–158 10.8 Integration by parts 240
C3 6.1 Separating fractions 148 10.9 A systematic approach
6.2 More partial fractions 152 to integration 246
1 Algebra and 3 Exponentials and Review 6 156 10.10 Volumes of revolution 248
functions 1–38 logarithms 79–90 Exit 6 158 Review 10 256
1.1 Combining algebraic fractions 2 3.1 The exponential function, e x
80
Exit 10 260
1.2 Algebraic division 4 3.2 The logarithmic function, ln x 84 7 Parametric
1.3 Mappings and functions 8 3.3 Equations involving e x and ln x 86 equations 159–178 11 Differential
1.4 Inverse functions 16 Review 3 88 7.1 Parametric equations and equations 261–274
1.5 The modulus function 22 Exit 3 90 curve sketching 160 11.1 First-order differential equations 262
1.6 Solving modulus equations 7.2 Points of intersection 166 11.2 Applications of differential
and inequalities 26 4 Differentiation 91–120 7.3 Differentiation 168 equations 268
1.7 Transformations of graphs 4.1 Trigonometric functions 92 7.4 Integration 172 Review 11 272
of functions 28 4.2 The exponential function, e x 98 Review 7 176 Exit 11 274
Review 1 34 4.3 The logarithmic function, ln x 102 Exit 7 178
Exit 1 38 4.4 The product rule 104
12 Vectors 275–296
4.5 The quotient rule 106 8 The binomial series 179–192 12.1 Basic definitions and notations 276
2 Trigonometry 39–74 4.6 The chain rule 110 8.1 The binomial series 180 12.2 Applications in geometry 282
2.1 Reciprocal trigonometric functions 40 4.7 Further applications 114 8.2 Using partial fractions 184 12.3 The scalar (dot) product 286
2.2 Trigonometric equations Review 4 118 8.3 Approximations 186 12.4 The vector equation of a
and identities 46 Exit 4 120 Review 8 190 straight line 290
2.3 Inverse trigonometric functions 50 Exit 8 192 Review 12 294
2.4 Compound angle formulae 54 5 Numerical methods 121–138 Exit 12 296
2.5 Double angle and half angle 5.1 Graphical methods 122 9 Differentiation 193–212
formulae 60 5.2 Iterative methods 130 9.1 Differentiating implicit functions 194 Revision 4 297–306
2.6 The equivalent forms for Review 5 136 9.2 Differentiating parametric functions 198
a cos + b sin 66 Exit 5 138 9.3 Growth and decay 200 Answers 307
Review 2 70 9.4 Rates of change 206 Index 345
Exit 2 74 Revision 2 139–146 Review 9 210
Exit 9 212
CD-ROM
Revision 1 75–78
Revision 3 213–216 The factor formulae 1–3
Formulae to learn
10 Integration 217–260 Formulae given in examination
10.1 The trapezium rule 218 Glossary
10.2 Integration as summation 222 C3 Practice paper
10.3 Integration using standard forms 224 C4 Practice paper
10.4 Further use of standard forms 226
10.5 Integration by substitution 228
10.6 Integration using trigonometric
identities 232
10.7 Integration using partial fractions 238
4. 1
Algebra and functions
This chapter will show you how to
combine algebraic fractions
understand that some mappings are also functions
find and use composite functions and inverse functions
use the modulus function and sketch graphs involving it
transform graphs of functions using translations, reflections,
stretches and combinations of these.
Before you start
You should know how to: Check in:
1 Combine numerical fractions. 1 Calculate
e.g. Calculate 3 + 1 = 9 + 2 = 11 a 3+2
C3
4 6 12 12 12 8 3
3 1 3 1
× = =
4 6 24 8 b 3×2
8 3
2 Find important facts to help you sketch 2 Sketch the graphs of
graphs of functions. a y = x(x - 2)
e.g. The graph of f(x) = 1 passes through the
x−2 b y = x2 + x - 2
( )
point 0, − 1 , has a vertical asymptote, x = 2, and
2 c y = 1 +2
x
approaches the value 0 as x ® ± ¥
3 Translate, reflect and stretch the graph 3 Write the equation of the image of the
of y = f(x) graph of y = f(x) when
e.g. When the graph of y = x2 is translated a f(x) = x2 is reflected in the x-axis
+2 units parallel to the x-axis, its equation
b f(x) = x2 - 5x is reflected in the y-axis
becomes y = (x - 2)2
⎛ −5 ⎞
c f(x) = x2 + x is translated by ⎜ ⎟
⎝ 0⎠
d f(x) = x2 is stretched with scale factor 4
parallel to the y-axis.
1
5. 1 Algebra and functions
Dividing by a fraction is equivalent to multiplying by its reciprocal.
1.1 Combining algebraic fractions
EXAMPLE 3
x 2 − 2x ÷ x 2 − 4
Addition and subtraction Simplify
x − 2x − 3
2 2x − 6
You can add or subtract algebraic fractions when they have a This method is similar to adding or ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
common denominator. subtracting numerical fractions.
x 2 − 2x ÷ x 2 − 4 = x 2 − 2x × 2x − 6 Multiply the first fraction by the
+ x − 2x − 3 2x − 6 x 2 − 2x − 3 x 2 − 4
2
reciprocal of the divisor.
E.g. b + y = by + by = ay by bx
a x ay bx
by is the common denominator.
x(x − 2) 2(x − 3)
= ×
You should find and use the lowest common denominator to (x + 1)(x − 3) (x − 2)(x + 2)
keep the calculation as simple as possible. 2x
=
(x + 1)(x + 2)
EXAMPLE 1
Evaluate a 3+1 b 2 + x−2
8 6 x2 + x x2 − 1
••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
Exercise 1.1
1 Simplify
a 3 + 1 = 9 + 4 = 13 The lowest common denominator
3p2q pq
8 6 24 24 24 6x 2 y
is 24. a b 8ab × 3ac
2 2 c ÷ 2
3xyz c 4b 2r 4r
b Factorise the algebraic expressions first:
( b ) × a b− a
2
2 + x−2 = 2 x−2 x × x +1 a 2
x + 3 × x −1
+ d e f
x 2 + x x 2 − 1 x(x + 1) (x + 1)(x − 1) x2 − 1 x2 2
x + x − 2 x2 + x − 6
2
C3
C3
2(x − 1) x(x − 2) The lowest common denominator x ÷ x 2 + 2x
2
h 1 × x2 − 9 ÷ x + 3
2
= + g
x(x + 1)(x − 1) x(x + 1)(x − 1) is x(x + 1)(x - 1). x 2 − 5x + 6 x −4 x x − 3x x
= 2x − 2 + x − 2x
2
x(x + 1)(x − 1) 2 Simplify and express as single fractions.
y
= x 2− 2
2
a x+ b a+b c 1 − 1
y z b a ax bx
x(x − 1)
d 1 + 1 e 1 +1 f 1 − 1
xy 2 x 2 y x +1 x a −1 a +1
Multiplication and division 1 1
g 2 + a +1 h 2+ 1 i 3− 2
You do not need to have a common denominator when you The method is similar to (a + 1) x x +1
multiply or divide algebraic fractions. multiplying or dividing
numerical fractions. 4 +2 2 + 1 3+ 2
j k l
y −2 3 x − 1 x2 − 1 x x2 + x
You can only cancel factors which are common to both the Remember when cancelling
numerator and denominator. brackets to cancel the whole 2 − y 1
bracket and not just part of it. m 1 − 2 n o + 2 2
z + 1 z2 − z − 2 y + 2 y + 3y + 2
2
x + 3x + 2 x + 4x + 3
2
3y y +1
EXAMPLE 2
p + q 2z − 2z r 2+ 1 − 1
Simplify x + 2 × x(x 2 − 1)
y2 − 4 y2 + y − 2 z + 2z − 3 z − 1
2
x 2 + 2x x 2 − 4
x2 − x x2 − x − 6
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3 Simplify
x + 2 × x(x 2 − 1) = x + 2 × x(x − 1)(x + 1) Factorise as much as you can
x2 − x x 2 − x − 6 x(x − 1) (x + 2)(x − 3) before cancelling. 1+ 1
a 1− x b y c x3 + 1 d x2 − 1 × x + 1
3
= x +1 1− 1 x2 − 1 x +x x −1
Cancel the fraction down to its 1
1− 2
x−3 simplest form. x y
2 3
6. 1 Algebra and functions
Some divisions result in a remainder.
1.2 Algebraic division
EXAMPLE 2
Divide a 4037 by 16
You can use long division to divide a polynomial of degree m The method is similar to dividing
by a polynomial of degree n, where m n. numbers using long division. b 4x3 + 3x + 7 by 2x - 1
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The degree of the resulting polynomial is m - n. a 252
16 4 0 3 7 )
3 2↓↓
EXAMPLE 1
Work out (x3 - 2x2 - x + 2) ÷ (x + 1) 83↓
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80↓
Set out as a long division: 37
32
)
x + 1 x 3 − 2x 2 − x + 2 Write each polynomial with the
highest power of x on the left. So 4037 ¸ 16 = 252 remainder 5
5 The remainder is 5.
The lead term of x + 1 is x. = 252 5
16
Divide the lead term x into x3 and write x2 in the answer space:
or 4037 = (252 ´ 16) + 5
Multiply this x2 by x + 1. x2 Write like terms in the
3 + x2 below and
x + 1 x 3 − 2x 2 − x + 2
Bring down the next
) same column. b 4x3 + 3x + 7 does not have an x2-term.
Insert a 0x2 to fill the place value for x2: The 0x2 is similar to the 0 in
Write x x3 + x2 ↓
subtract. term -x. the number 4037, which
−3x − x
2
2x 2 + x + 2
C3
C3
)
acts as place holder in the
2x − 1 4x 3 + 0x 2 + 3x + 7 hundreds column.
Repeat this cycle until the division is complete.
4x 3 − 2x 2 ↓ ↓
The next step is to divide the lead term x into -3x2 and write - 3x in the Compare this method with the 2x 2 + 3x ↓
numerical long division
answer space: 2x 2 − x ↓
4x + 7
x 2 − 3x + 2 243
4x − 2
x3 + x2 ↓ ↓
)
x + 1 x 3 − 2x 2 − x + 2 )
23 5 5 8 9
4 6↓ ↓
98↓
9 The remainder is 9.
−3x − x ↓
2 9 2↓ 4x 3 + 3x + 7 = 2x 2 + x + 2 remainder 9
So You can also write this result as
−3x − 3x ↓
2 69 2x − 1
Subtract: 69 4x3 + 3x + 7
2x + 2 0 quotient remainder = (2x - 1)(2x2 + x + 2) + 9
2x + 2 ¯ ¯
0 giving 5589 ÷ 23 = 243 You can expand these brackets
or 5589 = 243 ´ 23 = 2x + x + 2 + 9 2 to check your answer.
2x − 1
Since the remainder is 0, x + 1 is a factor of x3 - 2x2 - x + 2. Since the remainder is 0,
23 must be a factor of 5589. -
So (x3 - 2x2 - x + 2) ÷ (x + 1) = x2 - 3x + 2 divisor
You could also write this result as x3 - 2x2 - x + 2 = (x + 1) ´ (x2 - 3x + 2)
You can then expand these brackets to check your answer.
4 5