2. 3 About this book
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EXAMPLE 3
Divide 3x2 + 5 by x2 + 1
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Rewrite the numerator to involve a multiple of the denominator:
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through the solutions. 3x 2 + 5 = 3(x 2 + 1) + 2
x2 + 1 x2 + 1
Oxford University Press, at the address above Divide each term in the numerator:
=3+ 2
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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
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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