introduction this chapter focuses on basic manipulation of algebra it also goes over rules of...
TRANSCRIPT
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 FunctionsLike Terms
You can simplify expressions by collecting ‘like terms’
‘Like Terms’ are terms that are the same, for example;
5x and 3xb2 and -2b2
7ab and 8ab
are all ‘like terms’.
1A
Examplesa)
3 2 7 3 9x xy x xy
2x 5xy - 2
b)
2 23 6 4 2 3 3x x x x
2x - 3x + 1
c) 2 23( ) 2(3 4 )a b a b 23 3a b 2 6 8a b
23 11a b
Expand each bracket first
Algebra and FunctionsIndices (Powers)
You need to be able to simplify expressions involving Indices, where appropriate.
1B
m n m na a a m n m na a a
( )m n mna a1mmaa
1mma a
n n
mma a
4 2 63 3 3 7 3 45 5 5
2 4 8(6 ) 6
22
155
1
337 7
2 2
5510 10
Algebra and FunctionsIndices (Powers)
You need to be able to simplify expressions involving Indices, where appropriate.
1B
m n m na a a m n m na a a
( )m n mna a1mmaa
1mma a
n n
mma a
Examplesa)
2 5x x 7x
b)
2 32 3r r 56r
c) 4 4b b 0b1
d)
3 56 3x x 22x
e)
23 22a a82a6 22a a
f) 32 43x x6 427x x 227x
Algebra and FunctionsExpanding Brackets
You can ‘expand’ an expression by multiplying the terms inside the bracket by the term outside.
1C
Examplesa)
5(2 3)x 10 15x
b)
3 (7 4)x x 221 12x x
c) 2 3(3 2 )y y 2 53 2y y
d)
2 34 (3 2 5 )x x x x 2 3 412 8 20x x x
e)
2 (5 3) 5(2 3)x x x 210 6 10 15x x x
210 4 15x x
Algebra and FunctionsFactorising
Factorising is the opposite of expanding brackets. An expression is put into brackets by looking for common factors.
1D
3 9x a) 3( 3)x
Common Factor
32 5x xb) ( 5)x x x
28 20x xc) 4 (2 5)x x 4x2 29 15x y xyd) 3 (3 5 )xy x y 3xy
23 9x xye) 3 ( 3 )x x y 3x
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
+ 28+ 7x+ 7+ 4xx2x+ 4x
- 24- 8x- 8+ 3xx2x+ 3x
Algebra and Functions
x2 + 3x 2+
You get the last number in a Quadratic Equation by multiplying the 2 numbers in the brackets
You get the middle number by adding the 2 numbers in the brackets
(x + 2)(x + 1)
Algebra and Functions
x2 - 2x 15-
You get the last number in a Quadratic Equation by multiplying the 2 numbers in the brackets
You get the middle number by adding the 2 numbers in the brackets
(x - 5)(x + 3)
Algebra and Functionsx2 - 7x + 12
Numbers that multiply to give +
12+3 +4
-3 -4+12 +1
-12 -1
+6 +2
-6 -2
Which pair adds to give -7?
(x - 3)(x - 4)
So the brackets were originally…
Algebra and Functionsx2 + 10x +
16Numbers that
multiply to give + 16+1
+16-1 -16+2 +8
-2 -8+4 +4
-4 -4
Which pair adds to give +10?
(x + 2)(x + 8)
So the brackets were originally…
Algebra and Functionsx2 - x - 20
Numbers that multiply to give -
20+1 -20-1
+20+2 -10-2
+10+4 -5-4 +5
Which pair adds to give - 1?
(x + 4)(x - 5)
So the brackets were originally…
Algebra and FunctionsFactorising Quadratics
A Quadratic Equation has the form;
ax2 + bx + c
Where a, b and c are constants and a ≠ 0.
You can also Factorise these equations.
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
Examplesa)
2 6 8x x
The 2 numbers in brackets must: Multiply to give ‘c’ Add to give ‘b’
( 2)( 4)x x
Algebra and FunctionsFactorising Quadratics
A Quadratic Equation has the form;
ax2 + bx + c
Where a, b and c are constants and a ≠ 0.
You can also Factorise these equations.
1E
Examplesb)
2 4 5x x
The 2 numbers in brackets must: Multiply to give ‘c’ Add to give ‘b’
( 5)( 1)x x
Algebra and FunctionsFactorising Quadratics
A Quadratic Equation has the form;
ax2 + bx + c
Where a, b and c are constants and a ≠ 0.
You can also Factorise these equations.
1E
Examplesc) 2 25x
The 2 numbers in brackets must: Multiply to give ‘c’ Add to give ‘b’
( 5)( 5)x x
(In this case, b = 0)
This is known as ‘the difference of two squares’ x2 – y2 = (x + y)(x – y)
Algebra and FunctionsFactorising Quadratics
A Quadratic Equation has the form;
ax2 + bx + c
Where a, b and c are constants and a ≠ 0.
You can also Factorise these equations.
1E
Examplesd)
2 24 9x y
The 2 numbers in brackets must: Multiply to give ‘c’ Add to give ‘b’
(2 3 )(2 3 )x y x y
Algebra and FunctionsFactorising Quadratics
A Quadratic Equation has the form;
ax2 + bx + c
Where a, b and c are constants and a ≠ 0.
You can also Factorise these equations.
1E
Examplesd)
25 45x
The 2 numbers in brackets must: Multiply to give ‘c’ Add to give ‘b’ Sometimes, you need to
remove a ‘common factor’ first…
25( 9)x
5( 3)( 3)x x
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
+ 12+ 4x+ 4+ 3xx2x+ 3x
+ 12+ 8x+ 4+ 3x2x2x+ 32x
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
+3 -1
One of the values to the left will be doubled when the brackets are expanded
(2x + 1)(x - 3)
So the brackets were originally…
-6 +1-3 +2
+6 -1+3 -2 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
-11 -1
One of the values to the left will be doubled when the brackets are expanded
(2x + 11)(x + 1)
So the brackets were originally…
+22 +1+11 +2-22 -1-11 -2 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
-4 +1
+4 -1
One of the values to the left will be tripled when the brackets are expanded
(3x + 1)(x - 4)
So the brackets were originally…
+6 -2+2 -6-12 +1-4 +3 The -4 triples so it must
be on the opposite side to the ‘3x’
+12 -1+4 -3
Algebra and FunctionsExtending the rules of Indices
The rules of indices can also be applied to rational numbers (numbers that can be written as a fraction)
1F
m n m na a a m n m na a a
( )m n mna a1mmaa
1mma a
n n
mma a
Examplesa)
4 3x x 7x
b)
1 32 2x x
42x2x
c)2
3 3( )x233x
63x2x
d)
1.5 0.252 4x x 1.750.5x741
2x
Algebra and FunctionsExtending the rules of Indices
The rules of indices can also be applied to rational numbers (numbers that can be written as a fraction)
1F
m n m na a a m n m na a a
( )m n mna a1mmaa
1mma a
n n
mma a
Examplesa)
129 9
3
b)
1364 3 64
4
c)3249 3
49
343
d)
3225
32
1
25
3
1
25
1125
Algebra and FunctionsExtending the rules of Indices
The rules of indices can also be applied to rational numbers (numbers that can be written as a fraction)
1F
m n m na a a m n m na a a
( )m n mna a1mmaa
1mma a
n n
mma a
Examples
a)
123
123
32
b)
131
8
3
3
18
12
Algebra and FunctionsSurd Manipulation
You can use surds to represent exact values.
1G
ab a b
ab a b
a ab b
ExamplesSimplify the following…a) 12 4 3
2 3
b) 202
4 52
2 5
2
5
c) 5 6 2 24 294 2 4 6 49 65 6
4 6 7 65 6
8 6
Algebra and FunctionsRationalising
Rationalising is the process where a Surd is moved from the bottom of a fraction, to the top.
1H
ab
ab c
ab c
Multiply top and bottom by
Multiply top and bottom by
Multiply top and bottom by
b
b c
b c
ExamplesRationalise the following…
a)13
33
39
33
Algebra and FunctionsRationalising
Rationalising is the process where a Surd is moved from the bottom of a fraction, to the top.
1H
ab
ab c
ab c
Multiply top and bottom by
Multiply top and bottom by
Multiply top and bottom by
b
b c
b c
ExamplesRationalise the following…
b) 1
3 2
3 2
3 2
3 2
3 2 3 2
3 2
9 2 3 2 3 2
3 2
7
Algebra and FunctionsRationalising
Rationalising is the process where a Surd is moved from the bottom of a fraction, to the top.
1H
ab
ab c
ab c
Multiply top and bottom by
Multiply top and bottom by
Multiply top and bottom by
b
b c
b c
ExamplesRationalise the following…
c)
5 2
5 2
5 2
5 2
5 2 5 2
5 2 5 2
5 10 1025 10102
7 2 103
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!