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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)
0
1 2 3
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)
0
1 2 3
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
0
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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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