factorise this
(
1 of 73
)
Copy into
your notes
Words to learn
Factorise: put into brackets
Common Factor: numbers/letters in all terms
Variable: letters i.e. y
Quadratic: expression/equation with x2
Constant: number on its own
Coefficient: number before letter
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SLO
Simple Factorisation
(single brackets)
http://www.youtube.com/watch?v=iQYqR8OVECo
(easy single bracket)
3 of 73
http://www.youtube.com/watch?v=jqs0_9wO_0A&NR=1&feature=endscreen
(single bracket factorising)
Copy into
your notes
Factorising
Factorizing is the opposite of expanding.
Expanding or multiplying out
a(b + c)
ab + ac
Factorizing
4 of 73
Example 1:
Factorise 3a + 12b
Step 1 :
Find the highest number that divides into both
terms (common factor)? 3
Step 2 :
Find letters (variables) in common ?
None this time
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Step 3 :
Put common letters/numbers outside the bracket
3( a + 4b) = 3a + 12b
What do I need to
multiply 3 by to get 3a ?
What do I need to
multiply 3 by to get 12b ?
Step 4 :
Check: Expand brackets to find original question.
3(a + 4b) = 3a + 12b
6 of 73
Correct
Copy into
your notes
Example 2 : Factorise 14ab + 18ac
Step 1 :
Find the highest number that divides into both
terms? 2
Step 2 :
Find any letters in common? a
Step 3 :
Put the common letters/numbers ‘2a’ outside the
bracket
14ab + 18ac = 2a( 7b + 9c )
What do I need to multiply 2a
by to get 14ab ?
What do I need to multiply
2a by to get 18ac ?
Step 4 :
Check: Expand brackets to find original question.
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Factorization
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Equivalent expressions: find the pairs.
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Your Turn:
Factorise the following
Question
5x + 10
6a – 8
3x + x2
12n – 9n2
2p + 6p2 – 4p3
18a2 – 24ab
15x3 + 5x
10 of 73
Answer
5(x + 2)
2(3a – 4)
x(3 + x)
3n(4 – 3n)
2p(1 + 3p – 2p2)
6 a ( 3a  4b )
5 x (3 x  1)
2
Questions to do from the books
Achieve
Merit
Excellence
Gamma P18 Ex2.06
CAT 1.2
P32 Q215, 216,218
There are only three questions in the CAT book on this. Use next slide
for more questions.
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Your Turn:
Factorise the following (and check by expanding):
1:
2:
3:
4:
5:
15 – 3x = 3(5 – x)
2a + 10 = 2(a + 5)
ab – 5a = a(b – 5)
a2 + 6a = a(a + 6)
8x2 – 4x = 4x(2x – 1)
6: 10pq + 2p = 2p(5q + 1)
7: 20xy – 16x = 4x(5y - 4)
8: 24ab + 16a2 = 8a(3b + 2a)
9: r2 + 2 r =
r(r + 2)
10: 3a2 – 9a3 = 3a2(1 – 3a)
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SLO
Factorization by pairing
13 of 73
http://www.youtube.com/watch?v=KU6rNWjIZB4
(factorise by pairing)
Copy into
your notes
Factorization by pairing
Some expressions containing four terms can be factorized by
regrouping the terms into pairs that share a common factor. E.g.
Factorize 4a + ab + 4 + b
Step 1:
Split the 4 terms into 2 groups that share a common factor.
Step 2:
4a + ab + 4 + b = 4a + 4 + ab + b
Factorise in two parts.
4a + 4 + ab + b = 4(a + 1) + b(a + 1)
Step 3:
Rewrite as double brackets if there is a common factor.
4(a + 1) and + b(a + 1) share a common factor of (a + 1) so
we can write this as (a + 1)(4 + b)
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E.g. Factorize xy + 6 + 2y + 3x
Step 1:
Split the 4 terms into 2 groups that share a common factor.
xy + 6 + 2y + 3x = xy + 2y + 3x + 6
Step 2:
Factorise in two parts
xy + 2y + 3x + 6 = y(x + 2) + 3(x + 2)
Step 3:
Rewrite as double brackets if there is a common factor.
y(x + 2) and 3(x + 2) share a common factor of (x + 2) so we
can write this as (x + 2)(y + 3)
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E.g. Factorize 6ab + 9ad – 2bc – 3cd
Step 1:
Split the 4 terms into 2 groups.
6ab + 9ad – 2bc – 3cd
(no change!)
Step 2:
Factorise in two parts
3a(2b + 3d) – c(2b + 3d)
Step 3:
Rewrite as double brackets.
(3a – c)(2b + 3d).
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When we take
out a factor of
–c, –3cd
becomes + 3d
Your Turn:
Factorise the following
Question
wx + xz + wy + yz
6wx + 2xz + 3wy + yz
5wx + xz + 10wy + 2yz
6fh + 10fi + 9gh + 15gi
2wx – 2xz – wy + yz
8fh – 20fi + 6gh – 15gi
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Answer
(x + y)(w + z)
(2x + y)(3w + z)
(x + 2y)(5w + z)
(2f + 3g)(3h + 5i)
(2x – y)(w – z)
(4f + 3g)(2h – 5i)
Questions to do from the books
Achieve
Merit
Gamma
CAT 1.2
18 of 73
P32 Q217, 219 – 223
Excellence
SLO
Fractorising quadratics
http://www.youtube.com/watch?v=yfiMho1_t4k (easy quadratic)
http://www.youtube.com/watch?v=eF6zYNzlZKQ (harder quadratic)
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Quick challenge to help you later
We can write the problem: Find two numbers that add
to 10 and multiply to 21 in the diagram below?
multiply
21
3
add
20 of 73
7
10
x
Your Turn:
+
1)
+
2)
20
4
6
5
-9
-6
8
-6
6)
-2
6
-4
-13
-16
5)
-7
21 of 73
5
36
11
6
-1
3)
30
9
4)
x
2
-3
-1
Your Turn:
Find the two numbers that…
Multiply to give Add to give
12
8
12
7
35
12
-50
-5
10
-7
32
-12
-2
1
22 of 73
Answer
2 and 6
3 and 4
5 and 7
5 and -10
-2 and -5
-8 and -4
-1 and 2
Copy into
your notes
Quadratic expressions
Quadratics such as x2 + bx + c factorise into double brackets
i.e. (x + d)(x + e)
Expanding or multiplying out
a2 + 3a + 2
(a + 1)(a + 2)
Factorizing
23 of 73
Copy into
your notes
E.g. 1) Factorise x2 + 5x + 6
+
2
x
x
+ 5x + 6
The two numbers
add to 5
The two numbers
multiply to 6
The two numbers are +2 and +3
Therefore x2 + 5x + 6 = (x + 2)(x
24 of 73
+3
)
Copy into
your notes
E.g. 2) Factorise x2 – x – 6
+
2
x
x
–x – 6
The two numbers
add to -1
The two numbers
multiply to -6
The two numbers are +2 and -3
Therefore x2 – x – 6 = (x + 2)(x – 3)
25 of 73
Your Turn:
Factorise the following
Question
Answer
a  5a  6 (a  2) (a  3)
2
b  7b  12 (b  3) (b  4)
2
d  16d  15
(d  1)(d  15)
x  11x  18
( x  2)( x  9)
(a  1) (a  6)
2
2
a  7a  6
2
t  8t  12 (t  2)(t  6)
2
y  9 y  18 ( y  3)( y  6)
2
26 of 73
Your Turn:
Factorise the following
Question
Answer
p  13 p  30
( p  10)( p  3)
a  5a  6
(a  6) (a  1)
m2  4m  12
(m  6) (m  2)
2
2
k 2  7k  30 (k  10)(k  3)
2
w  3w  28 (w  7)(w  4)
2
( p  8)( p  3)
p  5 p  24
g2 – g – 20
(g – 5)(g + 4)
27 of 73
Factorizing quadratic expressions: Achieve
28 of 73
Matching quadratic expressions: Achieve
29 of 73
Questions to do from the books
Achieve
Gamma P19 Ex2.07
CAT 1.2
30 of 73
P32 Q224–244
Merit
Excellence
SLO
Quadratics with a common factor
http://www.youtube.com/watch?v=OMo98cTadOg&feature=related
(Quadratic with common factor)
31 of 73
Copy into
your notes
E.g. 1) Factorise 10x2 + 50x + 60
Take common factor out of all three terms
10x2 + 50x + 60 = 10[x2 + 5x + 6]
Factorise inside bracket
+
2
10[x
x
+ 5x + 6]
The two numbers
add to 5
The two numbers
multiply to 6
The two numbers are +2 and +3
Therefore 10x2 + 50x + 60 =10[(x + 2)(x + 3)]
32 of 73
Copy into
your notes
E.g. 2) Factorise 2x2 – 2x – 12
Take common factor out of all three terms
2x2 – 2x – 12 = 2[x2 – x – 6]
Factorise inside bracket
+
2
2[x
The two numbers
add to -1
x
–x – 6]
The two numbers
multiply to -6
The two numbers are +2 and -3
Therefore 2x2 – 2x – 12 = 2(x + 2)(x – 3)
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Your Turn:
Factorise the following
Question
10x2 + 40x + 30
5x2 - 15x + 10
2x2 + 14x - 60
4x2 - 16x - 48
3x2 + 21x + 30
5g2 – 5g – 100
4a2 + 12a + 8
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Answer
10(x + 3)(x + 1)
5(x – 2)(x – 1)
2(x + 10)(x – 3)
4(x + 2)(x – 6)
3(x + 2)(x + 5)
5(g – 5)(g + 4)
4(a + 1)(a + 2)
Questions to do from the books
Achieve
Gamma P21 Ex2.09
CAT 1.2
35 of 73
P35 Q257–262
Merit
Excellence
SLO
Merit: Factorising quadratics
http://www.youtube.com/watch?v=wah45uU8EAE
(quadratic factorising at MERIT level)
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Merit Quadratic factorising
Merit quadratics have a coefficient of x2 other than 1 i.e. the
number in front of x2 will not be one. There will also not be a
common factor. E.g.
2
3x
+ 17x + 20
Similar to Achieve questions, we still use 2 brackets but …
The two left hand numbers in the brackets multiply to give the coefficient of x 2
(
)(
) = 3x2 + 17x + 20
The two right hand numbers in the brackets multiply to give the constant
AND…
There might be many possible answers to the above but only
one will be correct i.e. generate the correct coefficient of x.
37 of 73
Copy into
your notes
E.g. 1) Factorise 3x2 + 17x + 20
x
2
3x
x
+ 17x + 20
Find two numbers that
multiply to give 3
The coefficients of x
must be 1 and 3
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Find two numbers that
multiply to give 20
The constants could be
(1,20), (20,1), (2,10),
(10,2), (4,5), (5,4)
Continued on next slide
Copy into
your notes
The coefficients of x
must be 1 and 3
From previous slide
The constants could be (1,20),
(20,1), (2,10), (10,2), (4,5), (5,4)
This leaves us with 6 possible answers. Only one is correct!
Possible answer Yes or No
(1x + 1)(3x + 20)
No
(1x + 20)(3x + 1)
No
(1x + 2)(3x + 10)
No
(1x + 10)(3x + 2)
No
(1x + 4)(3x + 5)
YES
(1x + 5)(3x + 4)
No
Notice at Merit
level that where
the 4 and 5 are
placed is
important
Therefore 3x2 + 17x + 20 = (x + 4)(3x + 5)
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A slightly quicker method
There is no quick method to factorise Merit quadratics as there
is with easier questions. You can either expand each possible
answer one at a time or use smiley face method below.
(dx + e)(fx + g)
=(dg + ef)x
Putting the above in words: follow the arrows and Multiply. Add
your 2 answers together and it will give the coefficient of x. E.g.
3x2 + 17x + 20 = (x + 4)(3x + 5)
(1 x 5 + 4 x 3) = 17
40 of 73
Factorizing quadratic expressions: Merit
41 of 73
Matching quadratic expressions: Merit
42 of 73
Your Turn:
Factorise the following
Question
25t2 – 20t + 4
4y2 + 12y + 5
8t2 – 2t – 1
6x2 + 11x – 10
43 of 73
Answer
(5t – 2)(5t – 2)
(2y + 1)(2y + 5)
(4t + 1)(2t – 1)
(3x – 2)(2x + 5)
Questions to do from the books
Achieve
Merit
Gamma
P22 Ex. 2.10/11
CAT 1.2
P35 Q263–274
44 of 73
Excellence
Difference of two squares
45 of 73
http://www.youtube.com/watch?v=_qyVzH3e1dY
(diff of 2 squares)
Factorizing the difference between two squares
A quadratic expression in the form
x2 – a2
is called the difference between two squares.
The difference between two squares can be factorized as
follows:
x2 – a2 = (x + a)(x – a)
For example,
9x2 – 16 = (3x + 4)(3x – 4)
25a2 – 1 = (5a + 1)(5a – 1)
m4 – 49n2 = (m2 + 7n)(m2 – 7n)
46 of 73
Your Turn:
Factorise the following
Question
x2 – 16
81x2 – 1
4x2 – 25
x2 – y2
16y2 – 64
¼ – t2
47 of 73
Answer
(x + 4)(x – 4)
(9x + 1)(9x – 1)
(2x + 5)(2x – 5)
(x + y)(x – y)
16(y + 2)(y – 2)
(½ + t)(½ – t)
Factorizing the difference between two squares
48 of 73
Matching the difference between two squares
49 of 73
Questions to do from the books
Achieve
Gamma P21 Ex2.08
CAT 1.2
50 of 73
P33 Q245–256
Merit
Excellence
Copy into
your notes
Summary of factorising
Single brackets: i.e. 6x2 + 9x = 3x(2x + 3)
Pairing: i.e. xy – 6 + 2y – 3x = xy + 2y – 3x – 6
= y(x + 2) – 3(x + 2)
= (x + 2)(y – 3)
Quadratics (a = 1): i.e.
x2
x
+
+ 3x + 2 = (x + 1)(x + 2)
Quadratics (common factor): i.e. 5x2 + 15x + 10 = 5(x2 + 3x + 2)
= 5(x + 1)(x + 2)
Merit Quadratics (a ≠ 1): i.e. 2x2 – 5x – 12 = (2x + 3)(x – 4)
Difference of two squares: i.e. 9x2 – 16 =
(3x + 4)(3x – 4)
51 of 73
Your Turn: Achieve Revision
1) Factorise
x  x  90  ( x  10)( x  9)
2
x 2  7 x  12  ( x  4)( x  3)
x 2  9 x  36  ( x  12)( x  3)
7) Factorise
5 x3  15 x  5 x( x 2  3)
4ab  8ac  12ad  4a(b  2c  3d )
4a(b – 2c + 3d)
x 2  100  ( x  10)( x  10)
2) Factorise
x 2  18 x  81  ( x  9) 2
1
1
1
x   ( x  )( x  )
4
2
2
9 x 2  100  (3 x  10)(3 x  10)
x 2  6 x  16  ( x  8)( x  2)
1
1
1
2
 x  (  x)(  x)
100
10
10
2
52 of 73
1) Factorise
x 2  5 x  36  ( x  9)( x  4)
x 2  11x  42  ( x  14)( x  3)
x 2  2 x  1  ( x  1) 2
1) Factorise
x 2  7 x  12  ( x  4)( x  3)
x 2  3x  28  ( x  7)( x  4)
x 2  1  ( x  1)( x  1)
1) Factorise
x 2  3x  70  ( x  10)( x  7)
x 2  17 x  72  ( x  9)( x  8)
x  100  ( x  10)( x  10)
2
53 of 73
Rational Expressions
(fractions)
http://www.youtube.com/watch?v=B4bVlDgHF5I
(Achieve rational expressions)
54 of 73
http://www.youtube.com/watch?v=9cetzdviKT4
(Merit rational expressions)
Copy into
your notes
Simplifying Fractions
Recap:
Probably the most widely known simplification is below. Can
you explain how it works
2
4
=
1
2
1𝑥2
2𝑥2
=
1
2
Rule:
Cross out common multiplications.
This works for letters as well……
55 of 73
Continued …..
Copy into
your notes
E.g. Simplify the following
5𝑎
7𝑎
3
56 of 73
2𝑥
1
=
4𝑥
2
2
5
=
7
5𝑡
1
=
15𝑡 3
1
1
12𝑡 2
3𝑡𝑡 3𝑡
=
=
4𝑡
1
1𝑡
1
3
Copy into
your notes
E.g. Simplify the following
5(𝑎 + 3)
5
=
7(𝑎 + 3)
7
1
2(𝑥 − 2) 2
=
𝑥−2
1
5(𝑡 + 8)(𝑡 − 4)
1(𝑡 − 4)
=
3 15(𝑡 + 8)
3
12(𝑡 − 5)2 3(𝑡 − 5)(𝑡 − 5) 3(𝑡 − 5)
=
=
1
1(𝑡 − 5)
1
4(𝑡 − 5)
3
57 of 73
Your Turn: Simplify the following
1)
4 x( x  7)
5( x  7)
5(𝑥 − 9)
3)
15(𝑥 + 3)
=
4x

5
(𝑥−9)
3(𝑥+3)
(𝑥 − 8)(𝑥 − 1)
5)
(𝑥 + 7)(𝑥 − 8)
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(𝑥 − 9)
1
2)
=
(𝑥 + 2)(𝑥 − 9)
(𝑥+2)
1
4)
=
3x 2 (2 x  3)
x

2 6 x(2 x  3)
2
(𝑥−1)
(𝑥+7)
Your Turn: Simplify the following
59 of 73
7)
3
3x  6 3( x  2)


4
4 x  8 4( x  2)
4
( x  2)( x  1)
( x  2)
x  x2


2
( x  3)( x  1)
( x  3)
x  4x  3
2
8)
6)
4 x  8 4( x  2)

x2
( x  2)
Your Turn: Factorise the following
5) Simplify
y  y 2 y(1  y) y


9)
3  3 y 3(1  y ) 3
2) Simplify
x 2  3 x  4 ( x  4)( x  1)

11) 2
x  7 x  12 ( x  4)( x  3)
x 1

x3
60 of 73
10)
x 2  3 x  4 ( x  1)( x  4)

2
x  6 x  8 ( x  4)( x  2)
x 1

x2
5) Simplify
x 2  6 x  7 ( x  1)( x  7)

12)
2
x  5 x  14 ( x  2)( x  7)
x 1

x2
Questions to do from the books
Achieve
Merit
Gamma P92 Ex7.04
P93 Ex7.05
CAT 1.2
P41 Q317–332
61 of 73
P41 Q311–316
Excellence