introduction this chapter focuses on basic manipulation of algebra it also goes over rules of...

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!