Algebraic Operations

1
Algebraic Operations
Factors / HCF
Common Factors
Difference of Squares
Factorising Trinomials (Quadratics)
Factor Priority
2
Starter Questions
S3 Credit
Q1. Multiply out (a) a (4y 3x) (b) (2x-1)(x4
)
Q2. True or false
www.mathsrevision.com
Q3. Write down all the number that divide into
12 without leaving a remainder.
3
Factors
S3 Credit
Using Factors
Learning Intention
Success Criteria

• To identify factors using factor pairs

• To explain that a factor divides into a number
  without leaving a remainder
• To explain how to find Highest Common Factors

• Find HCF for two numbers by comparing factors.

www.mathsrevision.com
4
Factors
S3 Credit
Factors
Example Find the factors of 56.
Numbers that divide into 56 without leaving a
remainder
F56 1 and 56
2 and 28
www.mathsrevision.com
4 and 14
7 and 8
5
Factors
S3 Credit
Highest Common Factor
Highest Common Factor
Largest Same Number
www.mathsrevision.com
We need to write out all factor pairs in order to
find the Highest Common Factor.
6
Factors
S3 Credit
Highest Common Factor
Example Find the HCF of 8 and 12.
F8 1 and 8 2 and 4
F12 1 and 12 2 and 6 3 and 4
www.mathsrevision.com
HCF 4
7
Factors
S3 Credit
Highest Common Factor
Example Find the HCF of 4x and x2.
F4x 1, and 4x
Fx2 1 and x2
2 and 2x
x and x
4 and x
HCF x
www.mathsrevision.com
Example Find the HCF of 5 and 10x.
F5 1 and 5
F10x 1, and 10x
2 and 5x
HCF 5
5 and 2x
10 and x
8
Factors
S3 Credit
Highest Common Factor
Example Find the HCF of ab and 2b.
F ab 1 and ab a and b
F2b 1 and 2b 2 and b
HCF b
www.mathsrevision.com
Example Find the HCF of 2h2 and 4h.
F 2h2 1 and 2h2 2 and h2 , h and 2h
F4h 1 and 4h 2 and 2h 4 and h
HCF 2h
9
Factors
S3 Credit
Find the HCF for these terms
8w

• (a) 16w and 24w
• 9y2 and 6y
• (c) 4h and 12h2
• (d) ab2 and a2b

3y
www.mathsrevision.com
4h
ab
10
Factors
S3 Credit
Now try Ex 2.1 3.1 First Column in each
Question Ch5 (page 86)
www.mathsrevision.com
11
Starter Questions
S3 Credit
Q1. Expand out (a) a (4y 3x) -2ay (b) (x
5)(x - 5)
Q2. Write out in full
www.mathsrevision.com
Q3. True or False all the factors of 5x2 are 1,
x, 5
12
Factorising
S3 Credit
Using Factors
Learning Intention
Success Criteria

• To identify the HCF for given terms.

• To show how to factorise terms using the Highest
  Common Factor and one bracket term.

• Factorise terms using the HCF and one bracket
  term.

www.mathsrevision.com
13
Check by multiplying out the bracket to get back
to where you started
Factorising
S3 Credit
Factorise 3x 15
Example
1. Find the HCF for 3x and 15
3
2. HCF goes outside the bracket
3( )
www.mathsrevision.com

• To see what goes inside the bracket
• divide each term by HCF

3x 3 x
15 3 5
3( x 5 )
14
Check by multiplying out the bracket to get back
to where you started
Factorising
S3 Credit
Factorise 4x2 6xy
Example
1. Find the HCF for 4x2 and 6xy
2x
2. HCF goes outside the bracket
2x( )
www.mathsrevision.com

• To see what goes inside the bracket
• divide each term by HCF

4x2 2x 2x
6xy 2x 3y
2x( 2x- 3y )
15
Factorising
S3 Credit
Factorise the following
3(x 2)

• (a) 3x 6
• 4xy 2x
• 6a 7a2
• (d) y2 - y

Be careful !
2x(2y 1)
www.mathsrevision.com
a(6 7a)
y(y 1)
16
Factorising
S3 Credit
Now try Ex 4.1 4.2 First 2 Columns only Ch5
(page 88)
www.mathsrevision.com
17
Starter Questions
S3 Credit
Q1. In a sale a jumper is reduced by 20. The
sale price is 32. Show that the original price
was 40
Q2. Factorise 3x2 6x
www.mathsrevision.com
Q3. Write down the arithmetic operation associate
d with the word difference.
18
Difference of Two Squares
S3 Credit
Learning Intention
Success Criteria

• Recognise when we have a difference of two
  squares.

• To show how to factorise the special case of the
  difference of two squares.

www.mathsrevision.com

• Factorise the difference of two squares.

19
Difference of Two Squares
S3 Credit
When an expression is made up of the difference
of two squares then it is simple to factorise
The format for the difference of two squares
www.mathsrevision.com
a2 b2
First square term
Second square term
Difference
20
Difference of Two Squares
S3 Credit
Check by multiplying out the bracket to get back
to where you started
a2 b2
First square term
Second square term
Difference
This factorises to
www.mathsrevision.com
( a b )( a b )
Two brackets the same except for and a -
21
Difference of Two Squares
S3 Credit
Keypoints
Format a2 b2
www.mathsrevision.com
Always the difference sign -
( a b )( a b )
22
Difference of Two Squares
S3 Credit
Factorise using the difference of two squares
(x 7 )( x 7 )

• (a) x2 72
• w2 1
• 9a2 b2
• (d) 16y2 100k2

( w 1 )( w 1 )
www.mathsrevision.com
( 3a b )( 3a b )
( 4y 10k )( 4y 10k )
23
Difference of Two Squares
S3 Credit
Trickier type of questions to factorise. Sometimes
we need to take out a common factor and then use
the difference of two squares.
Example
Factorise 2a2 - 18
2(a2 - 9)
First take out common factor
www.mathsrevision.com
Now apply the difference of two squares
2( a 3 )( a 3 )
24
Difference of Two Squares
S3 Credit
Factorise these trickier expressions.
6(x 2 )( x 2 )

• (a) 6x2 24
• 3w2 3
• 8 2b2
• (d) 27w2 12

3( w 1 )( w 1 )
www.mathsrevision.com
2( 2 b )( 2 b )
3(3 w 2 )( 3w 2 )
25
Difference of Two Squares
S3 Credit
Now try Ex 5.1 5.2 First 2 Columns only Ch5
(page 90)
www.mathsrevision.com
26
Starter Questions
S3 Credit
Q1. True or false y ( y 6 ) -7y y2 -7y 6
Q2. Fill in the ? 49 4x2 ( ? ?x)(? 2?)
www.mathsrevision.com
Q3. Write in scientific notation 0.0341
27
Factorising Using St. Andrews Cross method
S3 Credit
Learning Intention
Success Criteria

• Understand the steps of the St. Andrews Cross
  method.
• 2. Be able to factorise quadratics using SAC
  method.

• To show how to factorise trinomials ( quadratics)
  using
• St. Andrew's Cross method.

www.mathsrevision.com
28
Factorising Using St. Andrews Cross method
S3 Credit
There are various ways of factorising trinomials
(quadratics) e.g. The ABC method, FOIL method.
We will use the St. Andrews cross method to
factorise trinomials / quadratics.
www.mathsrevision.com
29
Removing Double Brackets
A LITTLE REVISION Multiply out the brackets and
Simplify
(x 1)(x 2)
1. Write down F O I L
x2
2x
x
2
x2 3x 2
2. Tidy up !
30
Factorising Using St. Andrews Cross method
We use the SAC method to go the opposite way
FOIL
(x 1)(x 2)
x2
3x
2
SAC
(x 1)(x 2)
x2
3x
2
31
Factorising Using St. Andrews Cross method
Strategy for factorising quadratics
Find two numbers that multiply to give last
number (2) and Diagonals sum to give middle
value 3x.
x2 3x 2
x
2
2
x
(2) x( 1) 2
1
x
1
x
(2x) ( 1x) 3x
( ) ( )
32
Factorising Using St. Andrews Cross method
Strategy for factorising quadratics
Find two numbers that multiply to give last
number (5) and Diagonals sum to give middle
value 6x.
x2 6x 5
x
5
5
x
(5) x( 1) 5
1
x
1
x
(5x) ( 1x) 6x
( ) ( )
33
One number must be and one -
Factorising Using St. Andrews Cross method
Strategy for factorising quadratics
Find two numbers that multiply to give last
number (-12) and Diagonals sum to give middle
value x.
x2 x - 12
x
4
4
x
(4) x( -3) -12
- 3
- 3
x
x
(4x) ( -3x) x
( ) ( )
34
Both numbers must be -
Factorising Using St. Andrews Cross method
Strategy for factorising quadratics
Find two numbers that multiply to give last
number (4) and Diagonals sum to give middle
value -4x.
x2 - 4x 4
x
- 2
- 2
x
(-2) x( -2) 4
- 2
- 2
x
x
(-2x) ( -2x) -4x
( ) ( )
35
One number must be and one -
Factorising Using St. Andrews Cross method
Strategy for factorising quadratics
Find two numbers that multiply to give last
number (-3) and Diagonals sum to give middle
value -2x
x2 - 2x - 3
x
- 3
- 3
x
(-3) x( 1) -3
1
x
1
x
(-3x) ( x) -2x
( ) ( )
36
Factorising Using St. Andrews Cross method
S3 Credit
Factorise using SAC method
(m 1 )( m 1 )

• (a) m2 2m 1
• y2 6y 5
• b2 b - 2
• (d) a2 5a 6

( y 5 )( y 1 )
www.mathsrevision.com
( b - 2 )( b 1 )
( a - 3 )( a 2 )
37
Factorising Using St. Andrews Cross method
S3 Credit
Now try Ex6.1 Ch5 (page 93)
www.mathsrevision.com
38
Starter Questions
S3 Credit
Q1. Cash price for a sofa is 700. HP terms are
10 deposit the 6 months equal payments of 120.
Show that you pay 90 using HP terms.
www.mathsrevision.com
Q2. Factorise 2 x x2
39
Factorising Using St. Andrews Cross method
S3 Credit
Learning Intention
Success Criteria

• Be able to factorise trinomials / quadratics
  using SAC.

• To show how to factorise trinomials ( quadratics)
  of the form ax2 bx c using SAC.

www.mathsrevision.com
40
One number must be and one -
Factorising Using St. Andrews Cross method
Strategy for factorising quadratics
Find two numbers that multiply to give last
number (-4) and Diagonals sum to give middle
value -x
3x2 - x - 4
3x
3x
- 4
- 4
(-4) x( 1) -4
1
x
1
x
(3x) ( -4x) -x
( ) ( )
41
One number must be and one -
Factorising Using St. Andrews Cross method
Strategy for factorising quadratics
Find two numbers that multiply to give last
number (-3) and Diagonals sum to give middle
value -x
2x2 - x - 3
2x
2x
- 3
- 3
(-3) x( 1) -3
1
x
1
x
(-3x) ( 2x) -x
( ) ( )
42
one number is and one number is -
Factorising Using St. Andrews Cross method
Two numbers that multiply to give last number
(-3) and Diagonals sum to give middle value (-4x)
4x2 - 4x - 3
4x
Factors 1 and -3 -1 and 3
Keeping the LHS fixed
x
Can we do it !
( ) ( )
43
Factorising Using St. Andrews Cross method
Find another set of factors for LHS
4x2 - 4x - 3
Repeat the factors for RHS to see if it
factorises now
2x
2x
- 3
- 3
Factors 1 and -3 -1 and 3
2x
2x
1
1
( ) ( )
44
Both numbers must be
Factorising Using St. Andrews Cross method
Find two numbers that multiply to give last
number (15) and Diagonals sum to give middle
value (22x)
8x222x15
8x
Keeping the LHS fixed
Factors 1 and 15 3 and 5
Find all the factors of (15) then try and
factorise
x
Can we do it !
( ) ( )
45
Factorising Using St. Andrews Cross method
Find another set of factors for LHS
8x222x15
Repeat the factors for RHS to see if it
factorises now
4x
4x
5
5
Factors 3 and 5 1 and 15
2x
2x
3
3
( ) ( )
46
Factorising Using St. Andrews Cross method
S3 Credit
Now try Ex 7.1 First 2 columns only Ch5 (page 95)
www.mathsrevision.com
47
Starter Questions
S3 Credit
Q1.
Use a multiplication table to expand out (2x
5)(x 5)
Q2. After a 20 discount a watch is on sale for
240. What was the original price of the watch.
www.mathsrevision.com
Q3. True or false 3a2 b ab2 a2b2(3b a)
48
Summary of Factorising
S3 Credit
Learning Intention
Success Criteria

• Be able use the factorise priorities to factorise
  various expressions.

• To explain the factorising priorities.

www.mathsrevision.com
49
Summary of Factorising
S3 Credit
When we are asked to factorise there is priority
we must do it in.

• Take any common factors out and put them
• outside the brackets.

2. Check for the difference of two squares.
www.mathsrevision.com
3. Factorise any quadratic expression left.
50
Summary of Factorising
S3 Credit
St. Andrews Cross method
2
squares
Difference
www.mathsrevision.com
Take Out Common Factor
51
If you can successfully complete this exercise
then you have the necessary skills to pass the
algebraic part of the course.
Summary of Factorising
S3 Credit
Now try Ex 8.1 Ch5 (page 97)
www.mathsrevision.com