1. Created by Mr. Lafferty@mathsrevision.com
Algebraic Operations
Factors / HCF
Common Factors
Difference of Squares
Factorising Trinomials (Quadratics)
Factor Priority
See Quadratic Theory for Exam Questions
3-Dec-17
Created by Mr.

2. Created by Mr. Lafferty@mathsrevision.com
Starter Questions
Nat 5
Q1. Remove the brackets
(a) a (4y – 3x) (b) (2x-1)(x+4)
Q2. Calculate
Q3. Write down all the number that divide
into 12 without leaving a remainder.
3-Dec-17
Created by Mr.

3. Created by Mr. Lafferty@www.mathsrevision.com
Factorising
Nat 5
Using Factors
Learning Intention
Success Criteria
We are learning how to factorise terms using the Highest Common Factor and one bracket term.
To identify the HCF for given terms.
Factorise terms using the HCF and one bracket term.
3-Dec-17
Created by Mr.

4. Factorising Example Factorise 3x + 15 1. Find the HCF for 3x and 15 3
Check by multiplying out the bracket to get back to where you started
Factorising
Nat 5
Example
Factorise 3x + 15
1. Find the HCF for 3x and 15 3
2. HCF goes outside the bracket
3( )
To see what goes inside the bracket
divide each term by HCF
3x ÷ 3 = x
15 ÷ 3 = 5
3( x + 5 )
3-Dec-17
Created by Mr.

5. Factorising Example Factorise 4x2 – 6xy
Check by multiplying out the bracket to get back to where you started
Factorising
Nat 5
Example
Factorise 4x2 – 6xy
1. Find the HCF for 4x2 and 6xy 2x
2. HCF goes outside the bracket
2x( )
To see what goes inside the bracket
divide each term by HCF
4x2 ÷ 2x =2x
6xy ÷ 2x = 3y
2x( 2x- 3y )
3-Dec-17
Created by Mr.

6. Created by Mr. Lafferty@mathsrevision.com
Factorising
Nat 5
Factorise the following :
3(x + 2)
(a) 3x + 6
4xy – 2x
6a + 7a2
(d) y2 - y
Be careful !
2x(2y – 1)
a(6 + 7a)
y(y – 1)
3-Dec-17
Created by Mr.

7. Factors Now try N5 TJ Ex 7.1 Ch7 (page 65) www.mathsrevision.com Nat 5
3-Dec-17
Created by Mr. Lafferty

8. Created by Mr. Lafferty@mathsrevision.com
Starter Questions
Nat 5
Q1. In a sale a jumper is reduced by 20%. The sale price is £32.
What is the original price of the jumper.
Q2. Factorise 3x2 – 6x
Q3. Write down the arithmetic operation
associated with the word ‘difference’.
3-Dec-17
Created by Mr.

9. Created by Mr. Lafferty@www.mathsrevision.com
Difference of
Two Squares
Nat 5
Learning Intention
Success Criteria
We are learning how to factorise the special case of the difference of two squares.
Recognise when we have a difference of two squares.
Factorise the difference of two squares.
3-Dec-17
Created by Mr.

10. Difference of Two Squares
Nat 5
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
a2 – b2
First
square term
Second
square term
Difference
3-Dec-17
Created by Mr.

11. Difference of Two Squares
Nat 5
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
( a + b )( a – b )
Two brackets the same except for + and a -
3-Dec-17
Created by Mr.

12. Always the difference sign -
Difference of
Two Squares
Nat 5
Keypoints
Format a2 – b2
Always the difference sign -
( a + b )( a – b )
3-Dec-17
Created by Mr. Lafferty

13. Difference of Two Squares
Nat 5
Factorise using the difference of two squares
(x + y )( x – y )
(a) x2 – y2
w2 – z2
9a2 – b2
(d) 16y2 – 100k2
( w + z )( w – z )
( 3a + b )( 3a – b )
( 4y + 10k )( 4y – 10k )
3-Dec-17
Created by Mr. Lafferty

14. Created by Mr. Lafferty@www.mathsrevision.com
Difference of
Two Squares
Nat 5
Now try N5 TJ
Ex 7.2 Q1
Ch7 (page 66)
3-Dec-17
Created by Mr.

15. Difference of Two Squares
Nat 5
Trickier type of questions to factorise.
Sometimes we need to take out a common
And the use the difference of two squares.
Example
Factorise 2a2 - 18
First take out common factor
2(a2 - 9)
Now apply the difference of two squares
2( a + 3 )( a – 3 )
3-Dec-17
Created by Mr. Lafferty

16. Difference of Two Squares
Nat 5
Factorise these trickier expressions.
6(x + 2 )( x – 2 )
(a) 6x2 – 24
3w2 – 3
8 – 2b2
(d) 27w2 – 12
3( w + 1 )( w – 1 )
2( 2 + b )( 2 – b )
3(3 w + 2 )( 3w – 2 )
3-Dec-17
Created by Mr. Lafferty

17. Created by Mr. Lafferty@www.mathsrevision.com
Difference of
Two Squares
Nat 5
Now try N5 TJ
Ex 7.2 Q2
Ch7 (page 66)
3-Dec-17
Created by Mr.

18. Starter Questions Q1. Multiple out the bracket and simplify.
Nat 5
Q1. Multiple out the bracket and simplify.
(a) y ( y + 6 ) -7y
Q2. Factorise 49 – 4x2
Q3. Write in scientific notation
3-Dec-17
Created by Mr.

19. Using St. Andrew’s Cross method
Factorising
Using St. Andrew’s Cross method
Nat 5
Learning Intention
Success Criteria
We are learning how to factorise trinomials ( quadratics) using
St. Andrew's Cross method.
Understand the steps of the St. Andrew’s Cross method.
2. Be able to factorise quadratics using SAC method.
3-Dec-17
Created by Mr.

20. Using St. Andrew’s Cross method
Factorising
Using St. Andrew’s Cross method
Nat 5
There various ways of factorising trinomials ( quadratics)
e.g. The ABC method, FOIL method.
We will use the
St. Andrew’s cross method
to factorise trinomials / quadratics.
3-Dec-17
Created by Mr.

21. Created by Mr. Lafferty@mathsrevision.com
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 !
3-Dec-17
Created by Mr.

22. Using St. Andrew’s Cross method
Factorising
Using St. Andrew’s Cross method
We use SAC method to go the opposite way
FOIL
(x + 1)(x + 2) x2
+ 3x
+ 2
SAC
(x + 1)(x + 2) x2
+ 3x
+ 2
3-Dec-17
Created by Mr.

23. Using St. Andrew’s Cross method
Factorising
Using St. Andrew’s 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 x
+ 2
+ 2
(+2) x( +1) = +2 x x
+ 1
+ 1
(+2x) +( +1x) = +3x
( ) ( )
3-Dec-17
Created by Mr.

24. Using St. Andrew’s Cross method
Factorising
Using St. Andrew’s 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 x
+ 5
+ 5
(+5) x( +1) = +5 x x
+ 1
+ 1
(+5x) +( +1x) = +6x
( ) ( )
3-Dec-17
Created by Mr.

25. Using St. Andrew’s Cross method
One number must be +
and one -
Factorising
Using St. Andrew’s 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 x
+ 4
+ 4
(+4) x( -3) = -12 x x
- 3
- 3
(+4x) +( -3x) = +x
( ) ( )
3-Dec-17
Created by Mr.

26. Using St. Andrew’s Cross method
Both numbers must be -
Factorising
Using St. Andrew’s 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 x
- 2
- 2
(-2) x( -2) = -4 x x
- 2
- 2
(-2x) +( -2x) = -4x
( ) ( )
3-Dec-17
Created by Mr.

27. Using St. Andrew’s Cross method
One number must be +
and one -
Factorising
Using St. Andrew’s 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 x
- 3
- 3
(-3) x( +1) = -3 x x
+ 1
+ 1
(-3x) +( x) = -2x
( ) ( )
3-Dec-17
Created by Mr.

28. Using St. Andrew’s Cross method
Factorising
Using St. Andrew’s Cross method
Nat 5
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 )
( b - 2 )( b + 1 )
( a - 3 )( a – 2 )
3-Dec-17
Created by Mr. Lafferty

29. Created by Mr. Lafferty@www.mathsrevision.com
Difference of
Two Squares
Nat 5
Now try N5 TJ
Ex 7.3 Q1 & 2
Ch7 (page 67)
3-Dec-17
Created by Mr.

30. Created by Mr. Lafferty@mathsrevision.com
Starter Questions
Nat 5
Q1. Cash price for a sofa is £700.
HP terms are 10% deposit the 6 months equal payments of £120.
How much more do you pay with HP.
Q2. Factorise 2 – 3x – x2
3-Dec-17
Created by Mr.

31. Using St. Andrew’s Cross method
Factorising
Using St. Andrew’s Cross method
Nat 5
Learning Intention
Success Criteria
To show how to factorise trinomials ( quadratics) of the form ax2 + bx +c using SAC.
Be able to factorise trinomials / quadratics using SAC.
3-Dec-17
Created by Mr.

32. Using St. Andrew’s Cross method
One number must be +
and one -
Factorising
Using St. Andrew’s 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 x x
+ 1
+ 1
(3x) +( -4x) = -x
( ) ( )
3-Dec-17
Created by Mr.

33. Using St. Andrew’s Cross method
One number must be +
and one -
Factorising
Using St. Andrew’s 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 x x
+ 1
+ 1
(-3x) +( +2x) = -x
( ) ( )
3-Dec-17
Created by Mr.

34. Using St. Andrew’s Cross method
one number is + and
one number is -
Factorising
Using St. Andrew’s Cross method
Two numbers that multiply to give
last number (-3)
and
Diagonals sum to give middle value (-4x)
4x2 - 4x - 3 4x
Keeping the LHS fixed
Factors
1 and -3
-1 and 3 x
Can we do it !
( ) ( )
3-Dec-17
Created by Mr.

35. Using St. Andrew’s Cross method
Factorising
Using St. Andrew’s 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
( ) ( )
3-Dec-17
Created by Mr.

36. Using St. Andrew’s Cross method
Both numbers must be +
Factorising
Using St. Andrew’s Cross method
Find two numbers that
multiply to give last number (+15)
and
Diagonals sum to give middle value (+22x)
8x2+22x+15 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 !
( ) ( )
3-Dec-17
Created by Mr.

37. Using St. Andrew’s Cross method
Factorising
Using St. Andrew’s Cross method
Find another set of
factors for LHS
8x2+22x+15
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
( ) ( )
3-Dec-17
Created by Mr.

38. Created by Mr. Lafferty@www.mathsrevision.com
Difference of
Two Squares
Nat 5
Now try N5 TJ
Ex 7.3 Q1 & Q3
Ch7 (page 68)
3-Dec-17
Created by Mr.

39. Created by Mr. Lafferty@mathsrevision.com
Starter Questions
Nat 5
Q1. Multiple out the brackets and simplify.
(a) ( 2x – 5 )( x + 5 )
Q2. After a 20% discount a watch is on sale
for £240.
What was the original price of the watch.
Q3. Factorise 3ab – b2
3-Dec-17
Created by Mr.

40. Created by Mr. Lafferty@www.mathsrevision.com
Summary of
Factorising
Nat 5
Learning Intention
Success Criteria
To explain the factorising priorities.
Be able use the factorise priorities to factorise various expressions.
3-Dec-17
Created by Mr.

41. Created by Mr. Lafferty@www.mathsrevision.com
Summary of
Factorising
Nat 5
When we are asked to factorise there is priority we
must do it in.
Only TWO terms
Take any common factors out and put them
outside the brackets.
2. Check for the difference of two squares.
3. Factorise any quadratic expression left.
THREE terms
3-Dec-17
Created by Mr.

42. Created by Mr. Lafferty@www.mathsrevision.com
If you can successfully complete this exercise then you have the necessary skills to pass the algebraic part of the course.
Difference of
Two Squares
Nat 5
Now try N5 TJ
Ex 7.3 Q4
Ch7 (page 68)
3-Dec-17
Created by Mr.