Download 1)_C1_Algebra_and_Functions

Introduction
• This chapter focuses on basic
manipulation of Algebra
• It also goes over rules of Surds and
Indices
• It is essential that you understand this
whole chapter as it links into most of
the others!
Algebra and Functions
Like Terms
You can simplify expressions by
collecting ‘like terms’
‘Like Terms’ are terms that are the
same, for example;
5x and 3x
b2 and -2b2
7ab and 8ab
are all ‘like terms’.
Examples
a)
3x  2 xy  7  x  3xy  9
2x  5xy - 2
b)
3x 2  6 x  4  2 x 2  3x  3
x 2 - 3x + 1
c)
3(a  b2 )  2(3a  4b2 )
3a  3b 2  6a  8b 2
Expand each
bracket first
3a  11b 2
1A
Algebra and Functions
Indices (Powers)
You need to be able to simplify expressions
involving Indices, where appropriate.
a m  a n  a mn
34  32  36
a m  a n  a mn
57  53  54
(a m )n  a mn
1
m
a  m
a
(62 )4  68
1
2
5  2
5
1
m
1
3
a ma
n
m
a 
 a
m
7 37
n
2
5
10 
 10 
5
2
1B
Algebra and Functions
Indices (Powers)
You need to be able to simplify expressions
involving Indices, where appropriate.
a m  a n  a mn
a a  a
m
n
mn
(a m )n  a mn
1
m
a  m
a
Examples
a) x 2  x5
b) 2r 2  3r 3  6r 5
c) b 4  b 4
e)
a 
 a
 a 3   2a 2
2
a 6  2a 2  2a 8
a ma
m
 b0
1
d) 6 x 3  3 x 5  2x 2
1
m
n
m
 x7
f)
n
3x 
2 3
 x4
27x 6  x 4  27x 2
1B
Algebra and Functions
Expanding Brackets
You can ‘expand’ an expression by
multiplying the terms inside the
bracket by the term outside.
Examples
a)
5(2 x  3)  10x  15
b)
3x(7 x  4)  21x 2  12 x
c)
y 2 (3  2 y3 )  3 y 2  2 y5
d)
4 x(3x  2 x 2  5x3 )
 12 x 2  8 x3  20 x 4
e)
2 x(5 x  3)  5(2 x  3)
 10 x 2  6 x  10 x  15
 10 x 2  4 x  15
1C
Algebra and Functions
Common
Factor
Factorising
Factorising is the opposite of
expanding brackets. An expression
is put into brackets by looking for
common factors.
a)
3x  9
3
 3( x  3)
b)
x2  5x
x
 x( x  5)
c)
8 x 2  20 x
4x
 4 x(2 x  5)
d)
9 x2 y  15xy 2
3xy
 3xy (3x  5 y )
e)
3x 2  9 xy
3x
 3 x( x  3 y )
1D
Algebra and Functions
• Expand the following pairs of
brackets
(x + 4)(x + 7)
 x2 + 4x + 7x + 28
 x2 + 11x + 28
(x + 3)(x – 8)
 x2 + 3x – 8x – 24
 x2 – 5x - 24
x
+4
x
x2
+ 4x
+7
+ 7x
+ 28
x
+3
x
x2
+ 3x
-8
- 8x
- 24
Algebra and Functions
(x + 2)(x + 1)
x2 +
You get the middle
number by adding the 2
numbers in the brackets
3x
+
2
You get the last number
in a Quadratic Equation
by multiplying the 2
numbers in the brackets
Algebra and Functions
(x - 5)(x + 3)
x2 -
You get the middle
number by adding the 2
numbers in the brackets
2x
-
15
You get the last number
in a Quadratic Equation
by multiplying the 2
numbers in the brackets
Algebra and Functions
x2 - 7x + 12
Numbers that
multiply to give + 12
Which pair adds to
give -7?
+3 +4
-3 -4
+12 +1
-12 -1
+6 +2
-6 -2
So the brackets were
originally…
(x - 3)(x - 4)
Algebra and Functions
x2 + 10x + 16
Numbers that
multiply to give + 16
Which pair adds to
give +10?
+1 +16
-1 -16
+2 +8
-2 -8
+4 +4
-4 -4
So the brackets were
originally…
(x + 2)(x + 8)
Algebra and Functions
x2 - x - 20
Numbers that
multiply to give - 20
Which pair adds to
give - 1?
+1 -20
-1 +20
+2 -10
-2 +10
+4 -5
-4 +5
So the brackets were
originally…
(x + 4)(x - 5)
Algebra and Functions
Factorising Quadratics
Examples
A Quadratic Equation has the
form;
a)
ax2
+ bx + c
Where a, b and c are constants and
a ≠ 0.
You can also Factorise these
equations.
x2  6 x  8
The 2 numbers in brackets must:
 Multiply to give ‘c’
 Add to give ‘b’
 ( x  2)( x  4)
REMEMBER
 An equation with an ‘x2’ in does
not necessarily go into 2 brackets.
You use 2 brackets when there are
NO ‘Common Factors’
1E
Algebra and Functions
Factorising Quadratics
Examples
A Quadratic Equation has the
form;
b)
ax2 + bx + c
Where a, b and c are constants and
a ≠ 0.
x2  4 x  5
The 2 numbers in brackets must:
 Multiply to give ‘c’
 Add to give ‘b’
 ( x  5)( x  1)
You can also Factorise these
equations.
1E
Algebra and Functions
Factorising Quadratics
Examples
A Quadratic Equation has the
form;
c)
ax2 + bx + c
Where a, b and c are constants and
a ≠ 0.
x 2  25
The 2 numbers in brackets must:
 Multiply to give ‘c’
 Add to give ‘b’ (In this case, b = 0)
 ( x  5)( x  5)
You can also Factorise these
equations.
This is known as ‘the
difference of two squares’
 x2 – y2 = (x + y)(x – y)
1E
Algebra and Functions
Factorising Quadratics
Examples
A Quadratic Equation has the
form;
d)
ax2 + bx + c
Where a, b and c are constants and
a ≠ 0.
4 x2  9 y 2
The 2 numbers in brackets must:
 Multiply to give ‘c’
 Add to give ‘b’
 (2 x  3 y )(2 x  3 y )
You can also Factorise these
equations.
1E
Algebra and Functions
Factorising Quadratics
Examples
A Quadratic Equation has the
form;
d)
ax2 + bx + c
Where a, b and c are constants and
a ≠ 0.
You can also Factorise these
equations.
5 x 2  45
The 2 numbers in brackets must:
 Multiply to give ‘c’
 Add to give ‘b’
 Sometimes, you need to remove
a ‘common factor’ first…
 5( x 2  9)
 5( x  3)( x  3)
1E
Algebra and Functions
• Expand the following pairs of
brackets
(x + 3)(x + 4)
 x2 + 3x + 4x + 12
 x2 + 7x + 12
(2x + 3)(x + 4)
 2x2 + 3x + 8x + 12
 2x2 + 11x + 12
x
+3
x
x2
+ 3x
+4
+ 4x
+ 12
2x
+3
x
2x2
+ 3x
+4
+ 8x
+ 12
When an x term has a ‘2’
coefficient, the rules are
different…
2 of the terms are
doubled
 So, the numbers in
the brackets add to
give the x term, WHEN
ONE HAS BEEN
DOUBLED FIRST
Algebra and Functions
2x2 - 5x - 3
Numbers that
multiply to give - 3
-3 +1
-6 +1
-3 +2
+3 -1
+6 -1
+3 -2
One of the values to the left
will be doubled when the
brackets are expanded
So the brackets were
originally…
(2x + 1)(x - 3)
The -3 doubles so it must
be on the opposite side
to the ‘2x’
Algebra and Functions
2x2 + 13x + 11
Numbers that
multiply to give + 11
+11 +1
+22 +1
+11 +2
-11 -1
-22 -1
-11 -2
One of the values to the left
will be doubled when the
brackets are expanded
So the brackets were
originally…
(2x + 11)(x + 1)
The +1 doubles so it must
be on the opposite side
to the ‘2x’
Algebra and Functions
3x2 - 11x - 4
Numbers that
multiply to give - 4
+2 -2
+6 -2
+2 -6
-4 +1
-12 +1
-4 +3
+4 -1
+12 -1
+4 -3
One of the values to the left
will be tripled when the
brackets are expanded
So the brackets were
originally…
(3x + 1)(x - 4)
The -4 triples so it must
be on the opposite side
to the ‘3x’
Algebra and Functions
Extending the rules of Indices
Examples
The rules of indices can also be applied
to rational numbers (numbers that can
be written as a fraction)
a)
a m  a n  a mn
a a  a
m
n
(a )  a
1
am  m
a
m n
mn
a 
 a
3
2
x x  x
4
2
 x2
c)
2
3 3
(x )  x
x
3
2
3
6
3
 x2
a ma
m
1
2
mn
1
m
n
m
b)
x 4  x 3  x 7
d)
n
2 x1.5  4 x 0.25  0.5x1.75
1 74
 x
2
1F
Algebra and Functions
Extending the rules of Indices
The rules of indices can also be applied
to rational numbers (numbers that can
be written as a fraction)
a a  a
m
n
mn
Examples
a)
b)
a m  a n  a mn
(a )  a
1
am  m
a
m n
mn
d)
a ma
a 
 a
m
9
 9
3
1
3
64  3 64
4
3
2
49 

49

3
 343
1
m
n
m
c)
1
2

25
3
2

1
25
3
2


1
25

3

1
125
n
1F
Algebra and Functions
Extending the rules of Indices
The rules of indices can also be applied
to rational numbers (numbers that can
be written as a fraction)
Examples
a)
2
 
3
1
a m  a n  a mn
a a  a
m
n
1
m
a ma
n
m
a 
 a
m

mn
(a m )n  a mn
1
am  m
a
n
1

2
 
3
b)
1
 
8
1
3
3
2
 31
  3 
 8
1

2
1F
Algebra and Functions
Surd Manipulation
You can use surds to represent
exact values.
Examples
Simplify the following…
a)
12  4  3
ab  a  b
ab  a b
a
a

b
b
2 3
b)
20
2

4 5
2

2 5
2
 5
5 6  2 24  294
 5 6 2 4 6  49 6
c)
 5 6 4 6 7 6
8 6
1G
Algebra and Functions
Rationalising
Examples
Rationalising is the process where a
Surd is moved from the bottom of a
fraction, to the top.
a
b
Multiply top and
bottom by
a
b c
Multiply top and
bottom by
a
b c
Multiply top and
bottom by
b
Rationalise the following…
a)
3
3
1


9
3
3

3
3
b c
b c
1H
Algebra and Functions
Rationalising
Examples
Rationalising is the process where a
Surd is moved from the bottom of a
fraction, to the top.
a
b
Multiply top and
bottom by
a
b c
Multiply top and
bottom by
b
Rationalise the following…
3 2


b)
3  2  3  2 
3 2


3  2 3  2 
1
b c
3  2 

a
b c
Multiply top and
bottom by
b c
9 2 3 2 3 2
3 2


7
1H
Algebra and Functions
Rationalising
Examples
Rationalising is the process where a
Surd is moved from the bottom of a
fraction, to the top.
a
b
Multiply top and
bottom by
a
b c
Multiply top and
bottom by
a
b c
Multiply top and
bottom by
b
b c
Rationalise the following…
c)




b c




2
5 2
5

2 




2
5 2
5

2
5 2
5 2
5
5
5  10 10 2
5  10 10 2
7  2 10
3
1H
Summary
• We have recapped our knowledge of
GCSE level maths
• We have looked at Indices, Brackets
and Surds
• Ensure you master these as they link
into the vast majority of A-level topics!