Algebraic Operations

1
Algebraic Operations
Removing Brackets
Pairs of Brackets
Factors
Common Factors
Difference of Squares
Factorising Trinomials (Quadratics)
Factor Priority
2
Starter Questions
Int 2
Q1. Calculate (a) -3 x 5 (b) -6 x -7
Q2. Calculate (a) w x w (b) -2a x 4a
www.mathsrevision.com
Q3. Find the gradient of the line if (3, 7)
and (12, 34)
3
Removing a Single Bracket
Int 2
Learning Intention
Success Criteria

1. To show how to multiply out (remove) a single
   bracket.

1. Understand the keypoints of multiplying out a
   expression with a single bracket.

1. Be able multiply out a expression with a single
   bracket.

www.mathsrevision.com
4
Removing a Single Bracket
Example 1
15
3(b 5)
3b
Example 2
- 8
4(w - 2)
4w
5
Removing a Single Bracket
Example 3
- a
a(y - 1)
ay
Example 4
- 6p
p(w - 6)
pw
6
Removing a Single Bracket
Example 5
3x
x(x 3)
x2
Example 6
- 6mq
3q(3q -2m)
9q2
7
Be careful with negatives !!
Removing a Single Bracket
Example 7
- 10
-2(h 5)
-2h
Example 8
9
-(g - 9)
-g
8
Removing a Single Bracket
Example 9
(x 4)
Find my Area
6
6(x 4)
6x
24
9
Be careful only multiply everything inside the
bracket
Removing a Single Bracket
Example 10
8 2(h 3)
8
6
2h
Now tidy up !
14
2h
10
Be careful only multiply everything inside the
bracket
Removing a Single Bracket
Example 10
-2(y - 1) 4
-2y
4
2
Now tidy up !
6
-2y
11
Be careful only multiply everything inside the
bracket
Removing a Single Bracket
Example 11
y - (4 - y)
y
y
- 4
Now tidy up !
- 4
12
Removing a Single Bracket
Example 12 Find the area of the picture frame.
(x 6)
(x 4)
x
4
x(x 6)
4(x 4)

Area
13
Removing a Single Bracket
Example 12
x(x 6) 4(x 4)
Area
x2
6x
- 4x
- 16
Now tidy up !
x2
2x
- 16
14
Removing a Single Bracket
Example 13
x(x - 3) 2(x - 3)
x2
- 3x
2x
- 6
Now tidy up !
x2
- x
- 6
15
Removing a Single Bracket
Int 2
Now try Exercise 1 Ch5 MIA (page 48)
www.mathsrevision.com
16
Starter Questions
Int 2
Q1. Calculate (a) -3y x 5y (b) -6q x (-4q)
Q2. Calculate (a) a(b - c) (b) -2a( b a)
www.mathsrevision.com
Q3. Write down the gradient and were the line
cuts the y axis. y 5 3x
17
Removing Double Brackets
Int 2
Learning Intention
Success Criteria

1. To show 2 methods for multiplying out brackets

1. Understand the keypoints of multiplying out
   double brackets.

1. Be able multiply out double brackets using 2
   methods.

www.mathsrevision.com
18
Removing Double Brackets
There two methods we can use to multiply out
DOUBLE brackets.
First Method
Simply remember the word
F
O
I
L
Multiply Last 2
Multiply First 2
Multiply Outside 2
Multiply Inside 2
19
Removing Double Brackets
Example 1 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 !
20
Removing a Single Bracket
Example 2 Multiply out the brackets and
Simplify
(x - 1)(x 2)
1. Write down F O I L
x2
2x
- x
- 2
x2 x - 2
2. Tidy up !
21
Removing Double Brackets
(x 1)(x - 2)
x2 - x - 2
(x - 1)(x - 2)
x2 - 3x 2
(x 3)(x 2)
x2 5x 6
(x - 3)(x 2)
x2 - x - 6
(x 3)(x - 2)
x2 x - 6
22
Removing a Single Bracket
Int 2
Now try Exercise 2 Q1 Ch5 MIA (page 50)
www.mathsrevision.com
23
Removing Double Brackets
We have Multiplication Table
Another way is to make a table
(y 2)(y 5)
y
2
y
5
Tidy up !
y2
5y
10
2y
y2 7y 10
24
Removing Double Brackets
Be careful with the negative signs
Example 2
(2x - 1)(x 3)
2x
- 1
x
3
Tidy up !
6x
2x2
2x2 5x - 3
-3
-x
25
Removing Double Brackets
Just a bigger Multiplication Table
Example 3
(x 4)(x2 3x 2)
x2
3x
2
x
4
Tidy up !
3x2
x3
2x
x3 7x2 14x 8
12x
8
4x2
26
Removing a Single Bracket
Int 2
Now try Exercise 2 Q3 Q5 Ch5 MIA (page 51)
www.mathsrevision.com
27
Starter Questions
Int 2
Q1. Remove the brackets (a) a (4y 3x)
(b) (2x-1)(x4)
Q2. Calculate The interest on 20 over 5 years
_at_ a compound interest of 7 per year.
www.mathsrevision.com
Q3. Write down all the number that divide into
12 without leaving a remainder.
28
Factors
Int 2
Using Factors
Learning Intention
Success Criteria

1. To identify factors using factor pairs

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

1. Find HCF for two numbers by comparing factors.

www.mathsrevision.com
29
Factors
Int 2
Factors
Example Find the factors of 56.
Always divide by 1 and find its pair
F56 1 and 56
From 2 find other factors and their pairs
www.mathsrevision.com
2 and 28 4 and 14 7 and 8
30
Factors
Int 2
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.
31
Factors
Int 2
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
32
Factors
Int 2
Highest Common Factor
Example Find the HCF of 4x and x2.
F4x 1, and 4x , 2 and 2x 4 and x
Fx2 1 and x2 x 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 , 5 and 2x 10 and
x
HCF 5
33
Factors
Int 2
Highest Common Factor
Example Find the HCF of ab and 2b.
F ab 1 and ab a and b
Fx2 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
34
Factors
Int 2
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
35
Factors
Int 2
Now try Exercise 3 Q3 and Q4 Ch5 (page 52)
www.mathsrevision.com
36
Starter Questions
Int 2
Q1. Remove the brackets (a) a (4y 3x)
(b) (x 5)(x - 5)
Q2. For the line y -x 5, find the
gradient and where it cuts the y axis.
www.mathsrevision.com
Q3. Find the highest common factor for p2q and
pq2.
37
Factorising
Int 2
Using Factors
Learning Intention
Success Criteria

1. To identify the HCF for given terms.

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

1. Factorise terms using the HCF and one bracket
   term.

www.mathsrevision.com
38
Check by multiplying out the bracket to get back
to where you started
Factorising
Int 2
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 )
39
Check by multiplying out the bracket to get back
to where you started
Factorising
Int 2
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 )
40
Factorising
Int 2
Factorise the following
3(x 2)

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

Be careful !
2x(y 1)
www.mathsrevision.com
a(6 7a)
y(y 1)
41
Factorising
Int 2
Now try Exercise 4 Start at Q2 Ch5 (page 53)
www.mathsrevision.com
42
Starter Questions
Int 2
Q1. Remove the brackets (a) a (8 3x 6a)
Q2. Factorise 3x2 6x
www.mathsrevision.com
Q3. Write down the first 10 square numbers.
43
Difference of Two Squares
Int 2
Learning Intention
Success Criteria

1. Recognise when we have a difference of two
   squares.

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

www.mathsrevision.com

1. Factorise the difference of two squares.

44
Difference of Two Squares
Int 2
When we have the special case that 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
45
Difference of Two Squares
Int 2
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 -
46
Difference of Two Squares
Int 2
Keypoints
Format a2 b2
www.mathsrevision.com
Always the difference sign -
( a b )( a b )
47
Difference of Two Squares
Int 2
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 )
www.mathsrevision.com
( 3a b )( 3a b )
( 4y 10k )( 4y 10k )
48
Difference of Two Squares
Int 2
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
2(a2 - 9)
First take out common factor
www.mathsrevision.com
Now apply the difference of two squares
2( a 3 )( a 3 )
49
Difference of Two Squares
Int 2
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 )
50
Difference of Two Squares
Int 2
Now try Exercise 5 Ch5 (page 54)
www.mathsrevision.com
51
Starter Questions
Int 2
Q1. Multiple out the brackets and
simplify. (a) ( y 3 )( y 6 )
Q2. Factorise 49 4x2
www.mathsrevision.com
Q3. Write down an equation parallel to y 4x 1
52
Factorising Using FOIL
Int 2
Learning Intention
Success Criteria

1. Be able to factorise quadratics using FOIL.

1. To show how to factorise trinomials ( quadratics)
   using FOIL.

www.mathsrevision.com
53
Factorising Using FOIL
Int 2
There various ways of factorising trinomials (
quadratics) e.g. The ABC method, St. Andrews
cross method. We will use our previous
knowledge and use the FOIL METHOD to factorise
quadratics.
www.mathsrevision.com
54
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 !
55
Factorising Using FOIL
We can also use FOIL to go the opposite way
FOIL
(x 1)(x 2)
x2
3x
2
FOIL
(x 1)(x 2)
x2
3x
2
56
Factorising Using FOIL
Strategy for factorising quadratics
FOIL
x2 3x2
x2
x x x
F OI L
2x
Put down two brackets
x
3x
( )( )
x
x
1
2
2
1 x 2
57
OI value will be (-1)x 12x 11x 1x (-12x)
-11x (-2x) 6x 4x 2x (-6x) -4x (-3x)
4x -x 3x (-4) -x
Factorising Using FOIL
Sometimes it can be trick to get OI correct
FOIL
x2 x - 12
x2
x x x
F OI L
4x
Put down two brackets
-3 x
x
( )( )
x
x
-3
4
-12
-3 x 4
58
Factorising Using FOIL
Int 2
Factorise using the difference of two squares
(m 1 )( m 1 )

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

( y 5 )( y 1 )
www.mathsrevision.com
( b - 2 )( b 1 )
( a - 3 )( a 2 )
59
Factorising Using FOIL
Int 2
Now try Exercise 6 Ch5 (page 56)
www.mathsrevision.com
60
Starter Questions
Int 2
Q1. Bacteria grows at a rate of 10 per
hour. Initially there was 600 bacteria in dish.
How many bacteria are there 5 hour later.
Q2. Find the volume of a cone with high 50cm and
diameter 10cm
www.mathsrevision.com
Q3. A line has gradient -7 and cuts the y
axis at -5. Write down the equation of the line.
61
Factorising Using FOIL
Int 2
Learning Intention
Success Criteria

1. Be able to factorise quadratics using FOIL.

1. To show how to factorise trinomials ( quadratics)
   using FOIL.

www.mathsrevision.com
62
OI value will be 12x (-1x) 11x (-12x) 1x
-11x 3x - 4x -x -3x 4x x
Factorising Using FOIL
Slightly harder example
FOIL
3x2 - x - 4
3x x x
3x2
F OI L
3x
Put down two brackets
- 4 x
- x
( )( )
- 4
3x
x
1
-4
-4 x 1
63
8x x x 8x2 or 4x x 2x 8x2
Factorising Using FOIL
Harder Still
FOIL
8x2 22 x 15
8x2
F OI L
Put down two brackets
22x
( )( )
15 x 1 15 or 3 x 5 15
15
64
Factorising Using FOIL
We just have to try all combinations to see what
works.
8x x x 8x2
4x x 2x 8x2
OI 22x
Middle term
121x
62x
23x
34x
29x
26x
43x
22x
65
Factorising Using FOIL
Int 2
Factorise using the difference of two squares
(m 1 )( m 1 )

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

( y 5 )( y 1 )
www.mathsrevision.com
( b - 2 )( b 1 )
( a - 3 )( a 2 )
66
Factorising Using FOIL
Int 2
Now try Exercise 7 Ch5 (page 57)
www.mathsrevision.com
67
Starter Questions
Int 2
Q1. Multiple out the brackets and
simplify. (a) ( 2x 5 )( x 5 )
Q2. Find the volume of a cylinder with high
6m and diameter 9cm
www.mathsrevision.com
Q3. Find the gradient and where line cut
y-axis. x y 1
68
Summary of Factorising
Int 2
Learning Intention
Success Criteria

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

1. To explain the factorising priorities.

www.mathsrevision.com
69
Summary of Factorising
Int 2
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.
70
If you can successfully complete this exercise
then you have the necessary skills to pass the
algebraic part of the course.
Summary of Factorising
Int 2
Now try Exercise 8 Ch5 (page 57)
www.mathsrevision.com