Breaking Brackets

What is Factorisation

Factorise 3a + 3b

5a + 5

Examples Interactive

Interactive

2d+8 3e+15

7g-21 Interactive

10a+5

12b-3

Interactive

4a+6 8d-12

Interactive

What is Factorisation

Factorisation is the opposite of multiplying.12 = 6 x 2

6 is a factor of 12 2 is a factor of 12

When you factorise you produce a multiplication sum12 = 1 x 12

12 = 2 x 6

12 = 3 x 4

The factorisations we normally do in Maths involve letters

3a = 3 x a

5f = 5 x f 7k = 7 x k

ab = a x b

Factorise 3a + 3b

Factorise 3a + 3b

a

a

a

3a

b

b

b

3b

a + ba + ba + b

Three rows with ( a + b ) in each row. The Maths code is 3 ( a + b )

3a + 3b = 3 ( a + b )

Factorise 7d + 7g

Factorise 7d + 7g

d

d

d

7d

g

g

g

7g

d + gd + gd + g

Seven rows with ( d + g ) in each row. The Maths code is 7 ( d + g )

7d + 7g

= 7 ( d + g )d g

d g

d g

d g

d + gd + gd + gd + g

Factorise 8c + 8d

Factorisation is the opposite of multiplying.In this case a term

==

8 x 8 x cc

==

8d8d

Take each term and put time signs between numbers and letters

++8c8c

xx (( ))……………………

by a single bracket

8c8c 8d8d

Find the HCF of both numbers

In this case both numbers are the same.

The HCF is 8.

++

From the previous examples you probably know that

8c + 8d =

A method follows8 ( c + d )

8 x 8 x dd

Factorisation Opposite of Multiplying a Bracket

8 c8 c ++ 8 d8 d

(( ))==

++

Put in brackets and one arrow for each term

Both terms contain 8 so the common factor is

x x88 88

8c = 8 8c = 8 xx

cccc

8d = 8 x8d = 8 x

8c =8 x c8c =8 x c8c8c

dddd

What do you multiply 8 by to get the two termsCopy down the sign

Put 8 at the front of the brackets

88 8c =8 x d8c =8 x d8d8d

More Examples

9 9 ( )( )5 5 ( )( )

22x x + 2y + 2y ==

xx + + yy

2 2 ( )( )

9p9p + 9q + 9q ==

p + p + qq

5a5a + 5e + 5e ==

a + a + ee

3 3 ( )( )

3s3s – 3t – 3t ==

s - ts - t

11 ( 11 ( ))

11y11y – 11t – 11t ==

y - ty - t

12 ( )12 ( )12g12g – 12w – 12w ==

g - g - ww

ExamplesExamples

(a) 7f(a) 7f + 7k+ 7k

(b) 2h(b) 2h + 2j+ 2j

(c) 5t(c) 5t – 5u– 5u

(d) 14g(d) 14g – 14k– 14k

(e) 26b(e) 26b + + 26v26v

FactoriseFactorise

=7(f=7(f + k + k ))=2(h=2(h + j + j ))=5(t=5(t - - u )u )=14(g=14(g - k - k ))=26(b=26(b + + v )v )

5a + 5b type Examples

a)a)

b)b)

c)c)

d)d)

e)e)

f)f)

g)g)

3r-3s3r-3s 3(r-s)3(r-s)

4r-4v4r-4v

9r+9w9r+9w

7r+7g7r+7g

2r-2h2r-2h

6s-6v6s-6v

8s+8w8s+8w

4(r-v)4(r-v)

9(r+w)9(r+w)

7(r+g)7(r+g)

2(r-h)2(r-h)

6(s-v)6(s-v)

8(s+w)8(s+w)

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4 5 6

7 8 9

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Factorise 5a + 5

5a5a ++ 55

(( ))==

++

Put in brackets and one arrow for each term

Both terms contain 5 so the common factor is

x x55 55

5a = 5 5a = 5 xx

aaaa

5 = 5 x5 = 5 x 1111

What do you multiply 5 by to get the two termsCopy down the sign

Put 5 at the front of the brackets

55

3a + 3 type Examples

a)a)

b)b)

c)c)

d)d)

e)e)

f)f)

g)g)

4b+44b+4 4(b+1)4(b+1)

8m+88m+8

2n-22n-2

6v+66v+6

9w+99w+9

3m+33m+3

7n+77n+7

8(m+1)8(m+1)

2(n-1)2(n-1)

6(v+1)6(v+1)

9(w+1)9(w+1)

3(m+1)3(m+1)

7(n+1)7(n+1)

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4 5 6

7 8 9

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How to Factorise 2d + 8

88++2 d2 d

(( ))==

++x x

Factorising is the opposite Factorising is the opposite of multiplying the of multiplying the

contents of a bracketcontents of a bracket

++

by a number ( or a term by a number ( or a term ))

Need to find a number that you can multiply something to get 2d and 8.

Find a common factor

How to Factorise 2d + 8

88++2 d2 d

(( ))==

++

xx

2222

++

2d is one term

8 is the 2nd term

Concentrate on the number part of each … 2 and 8

2d

2 x d

8

2 x 4

2 d 4

2 is a COMMON FACTOR of 2d and 8

In rows

d

d

2d

4

4

8

d + 4d + 4

2d + 8

= 2 ( d + 4 )

Two rows with ( d + 4 ) in each row.

The Maths code is 2 (d + 4 )

How to Factorise 3e + 15

1515++3e3e

(( ))==

++

xx

3333

++3 e 5

3 is a COMMON FACTOR of 3e and 15

Numbers 3 and 15

3 x 5 = 15

3 x e + 3 x 5

How to Factorise 7g - 21

2121--7g7g

(( ))==

--

xx

7777

--7 e 3

7 is a COMMON FACTOR of 7g and 21

Numbers 7 and 21

7 x 3 = 21

7 x g + 7 x 3

**** - between 7g and 21 - in brackets ***********

4a + 12 type Examples

a)a)

b)b)

c)c)

d)d)

e)e)

f)f)

g)g)

9w-819w-81

7p+287p+28

2q+122q+12

3y-213y-21

4z-324z-32

6p-306p-30

8q+168q+16

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4 5 6

7 8 9

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How to Factorise 10a + 5

55++10a10a

(( ))==x

x55

55

++5 2a 1

5 is a COMMON FACTOR of 10a and 5

Numbers 10 and 5

5 x 2 = 10

5 x 2 x a

+ 5 x 1

10a10a 55

5 x 2 = 10 …… don’t want 10 want 10a5 x 2a = 10a

How to Factorise 12b - 3

33--12b12b

(( ))==x

x33

33

--3 4b 1

3 is a COMMON FACTOR of 12b and 3

Numbers 12 and 3

3 x 4 = 12

3 x 4 x b

- 3 x 1

12b12b 33

3 x 4 = 12 …… don’t want 12 want 12b3 x 4b = 12b

12b - 3 type Examples

a)a)

b)b)

c)c)

d)d)

e)e)

f)f)

g)g)

45d+545d+5 5(9d+1)5(9d+1)

12m+412m+4

12n-612n-6

48e+848e+8

12f+312f+3

16m+216m+2

45n-945n-9

4(3m+1)4(3m+1)

6(2n-1)6(2n-1)

8(6e+1)8(6e+1)

3(4f+1)3(4f+1)

2(8m+1)2(8m+1)

9(5n-1)9(5n-1)

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4 5 6

7 8 9

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How to Factorise 4a + 6

66++4a4a

(( ))==x

x22

22

++2 2a 3

2 is a COMMON FACTOR of 4a and 6

Numbers 4 and 6

4 x = 6

2 x 2 x a

+ 2 x 3

4a4a 66

What times table are 4 and 6 both in2 x 2 = 4 2 x 3 = 6

How to Factorise 8d - 12

1212--8d8d

(( ))==x

x44

44

--4 2d 3

4 is a COMMON FACTOR of 8d and 12

Numbers 8 and 12

8 x = 12

4 x 2 x d

- 4 x 3

8d8d 1212

What times table are 8 and 12 both in2 x 4 = 8 2 x 6 = 12

4 x 2 = 8 4 x 3 = 12

No Multiple Examples

a)a)

b)b)

c)c)

d)d)

e)e)

f)f)

g)g)

40s+3040s+30 10(4s+3)10(4s+3)

12y+912y+9

10z-1510z-15

24g-1824g-18

25h-2025h-20

8y-208y-20

18z+2418z+24

3(4y+3)3(4y+3)

5(2z-3)5(2z-3)

6(4g-3)6(4g-3)

5(5h-4)5(5h-4)

4(2y-5)4(2y-5)

6(3z+4)6(3z+4)

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4 5 6

7 8 9

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