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Algebraic Operations
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Nat 5
Factors / HCF
Common Factors
Difference of Squares
Factorising Trinomials (Quadratics)
Factor Priority
See Quadratic Theory for Exam Questions
11-May-17
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Starter Questions
Nat 5
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Q1.
Remove the brackets
(a) a (4y – 3x)
(b)
(2x-1)(x+4)
Q2. Calculate
5 1

6 3
Q3. Write down all the number that divide
into 12 without leaving a remainder.
11-May-17
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Factorising
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Nat 5
Using Factors
Learning Intention
1. We are learning how to
factorise terms using the
Highest Common Factor
and one bracket term.
11-May-17
Success Criteria
1. To identify the HCF for
given terms.
2. Factorise terms using the
HCF and one bracket term.
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Check by multiplying
Factorising
Nat 5
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Example
out the bracket to get
back to where you
Factorise 3x +started
15
1.
Find the HCF for 3x and 15
2.
HCF goes outside the bracket
3.
To see what goes inside the bracket
divide each term by HCF
3x ÷ 3 = x
11-May-17
15 ÷ 3 = 5
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3
3(
)
3( x + 5 )
Check by multiplying
Factorising
Nat 5
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Example
out the bracket to get
back to where you
Factorise 4x2 –started
6xy
1.
Find the HCF for 4x2 and 6xy
2.
HCF goes outside the bracket
3.
To see what goes inside the bracket
divide each term by HCF
4x2 ÷ 2x =2x
11-May-17
6xy ÷ 2x = 3y
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2x
2x(
)
2x( 2x- 3y )
Factorising
Nat 5
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Factorise the following :
(a)
3(x + 2)
3x + 6
Be
careful !
2x(2y – 1)
(b)
4xy – 2x
(c)
6a + 7a2
a(6 + 7a)
(d)
y2 - y
y(y – 1)
11-May-17
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Factors
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Nat 5
Now try
N5 TJ Ex 7.1
Ch7 (page 65)
11-May-17
Created by Mr. Lafferty
Starter Questions
Nat 5
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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’.
11-May-17
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Difference of
Two Squares
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Nat 5
Learning Intention
1. We are learning how to
factorise the special case
of the difference of two
squares.
Success Criteria
1. Recognise when we have a
difference of two squares.
2. Factorise the difference of
two squares.
11-May-17
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Difference of
Two Squares
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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
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Difference
Second
square term
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Difference of
Two Squares
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Nat 5
2 by multiplying
a2 – out
bCheck
the bracket to get
First
square term
back to where you
Second
Difference started
square term
This factorises to
( a + b )( a – b )
Two brackets the same except for + and a 11-May-17
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Difference of
Two Squares
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Nat 5
Keypoints
Format
a2 – b2
Always the difference sign ( a + b )( a – b )
11-May-17
Created by Mr. Lafferty
Difference of
Two Squares
Nat 5
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Factorise using the difference of two squares
(a)
x2 – y2
(x + y )( x – y )
(b)
w2 – z2
( w + z )( w – z )
(c)
9a2 – b2
(d)
16y2 – 100k2
11-May-17
( 3a + b )( 3a – b )
( 4y + 10k )( 4y – 10k )
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Difference of
Two Squares
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Nat 5
Now try N5 TJ
Ex 7.2 Q1
Ch7 (page 66)
11-May-17
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Difference of
Two Squares
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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 )
11-May-17
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Difference of
Two Squares
Nat 5
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Factorise these trickier expressions.
(a)
6x2 – 24
6(x + 2 )( x – 2 )
(b)
3w2 – 3
3( w + 1 )( w – 1 )
(c)
8 – 2b2
(d)
11-May-17
27w2 – 12
2( 2 + b )( 2 – b )
3(3 w + 2 )( 3w – 2 )
Created by Mr. Lafferty
Difference of
Two Squares
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Nat 5
Now try N5 TJ
Ex 7.2 Q2
Ch7 (page 66)
11-May-17
Created by Mr. [email protected]
Starter Questions
Nat 5
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Q1.
Multiple out the bracket and simplify.
(a)
y ( y + 6 ) -7y
Q2. Factorise
49 – 4x2
Q3. Write in scientific notation 0.0341
11-May-17
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Factorising
Using St. Andrew’s Cross method
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Nat 5
Learning Intention
1. We are learning how to
factorise trinomials (
quadratics) using
St. Andrew's Cross method.
11-May-17
Success Criteria
1. Understand the steps of the
St. Andrew’s Cross method.
2. Be able to factorise
quadratics using SAC method.
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Factorising
Using St. Andrew’s Cross method
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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.
11-May-17
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Removing
Double Brackets
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Nat 5
A LITTLE REVISION
Multiply out the brackets and Simplify
(x + 1)(x + 2)
1.
Write down
2.
Tidy up !
11-May-17
F
O
I
L
x2 + 2x + x + 2
x2 + 3x + 2
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Factorising
Using St. Andrew’s Cross method
Nat 5
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We use SAC method to go the opposite way
FOIL
(x + 1)(x + 2)
(x + 1)(x + 2)
11-May-17
SAC
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x2 + 3x+ 2
x2 + 3x+ 2
Factorising
Using St. Andrew’s Cross method
Nat 5
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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
x
+1
(
)(
11-May-17
)
Created by Mr. [email protected]
(+2) x( +1) = +2
(+2x) +( +1x) = +3x
Factorising
Using St. Andrew’s Cross method
Nat 5
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Strategy for factorising quadratics
x2 + 6x + 5
Find two numbers that
multiply to give last number (+5)
and
Diagonals sum to give middle value +6x
x
+5
x
+1
(
11-May-17
)(
)
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(+5) x( +1) = +5
(+5x) +( +1x) = +6x
Factorising
One number
Using St. Andrew’s Cross method
must be +
and one -
Nat 5
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Strategy for factorising quadratics
x2 + x - 12
Find two numbers that
multiply to give last number (-12)
and
Diagonals sum to give middle value +x.
x
+4
x
-3
(
11-May-17
)(
)
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(+4) x( -3) = -12
(+4x) +( -3x) = +x
Factorising
Both numbers
Using St. Andrew’s Cross method
must be -
Nat 5
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Strategy for factorising quadratics
x2 - 4x + 4
Find two numbers that
multiply to give last number (+4)
and
Diagonals sum to give middle value -4x.
x
-2
x
-2
(
11-May-17
)(
)
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(-2) x( -2) = -4
(-2x) +( -2x) = -4x
Factorising
One number
Using St. Andrew’s Cross method
must be +
and one -
Nat 5
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Strategy for factorising quadratics
x2 - 2x - 3
Find two numbers that
multiply to give last number (-3)
and
Diagonals sum to give middle value -2x
x
-3
x
+1
(
11-May-17
)(
)
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(-3) x( +1) = -3
(-3x) +( x) = -2x
Factorising
Using St. Andrew’s Cross method
Nat 5
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Factorise using SAC method
(a)
m2 + 2m +1
(m + 1 )( m + 1 )
(b)
y2 + 6y + 5
( y + 5 )( y + 1 )
(c)
b2 – b - 2
( b - 2 )( b + 1 )
(d)
a2 – 5a + 6
( a - 3 )( a – 2 )
11-May-17
Created by Mr. Lafferty
Difference of
Two Squares
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Nat 5
Now try N5 TJ
Ex 7.3 Q1 & 2
Ch7 (page 67)
11-May-17
Created by Mr. [email protected]
Starter Questions
Nat 5
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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
11-May-17
2 – 3x – x2
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Factorising
Using St. Andrew’s Cross method
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Nat 5
Learning Intention
1. To show how to factorise
trinomials ( quadratics) of
the form ax2 + bx +c using
SAC.
11-May-17
Success Criteria
1. Be able to factorise
trinomials / quadratics
using SAC.
Created by Mr. [email protected]
Factorising
One number
Using St. Andrew’s Cross method
must be +
and one -
Nat 5
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Strategy for factorising quadratics
3x2 - x - 4
Find two numbers that
multiply to give last number (-4)
and
Diagonals sum to give middle value -x
3x
-4
x
+1
(
11-May-17
)(
)
Created by Mr. [email protected]
(-4) x( +1) = -4
(3x) +( -4x) = -x
Factorising
One number
Using St. Andrew’s Cross method
must be +
and one -
Nat 5
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Strategy for factorising quadratics
2x2 - x - 3
Find two numbers that
multiply to give last number (-3)
and
Diagonals sum to give middle value -x
2x
-3
x
+1
(
11-May-17
)(
)
Created by Mr. [email protected]
(-3) x( +1) = -3
(-3x) +( +2x) = -x
Factorisingone number is +
Using St. Andrew’s Cross method
and
one number is -
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Nat 5
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
(
11-May-17
)(
)
Can we do it !
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Factorising
Using St. Andrew’s Cross method
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Nat 5
Find another set of
factors for LHS
4x2 - 4x - 3
2x
-3
2x
+1
(
11-May-17
)(
Repeat the factors for
RHS to see if it
factorises now
)
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Factors
1 and -3
-1 and 3
Factorising
Both numbers
Using St. Andrew’s Cross method
must be +
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Nat 5
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 factors
Find
all the
3 andtry
5 and factorise
of (+15) then
x
(
11-May-17
)(
)
Can we do it !
Created by Mr. [email protected]
Factorising
Using St. Andrew’s Cross method
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Nat 5
Find another set of
factors for LHS
8x2+22x+15
4x
+5
2x
+3
(
11-May-17
)(
Repeat the factors for
RHS to see if it
factorises now
)
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Factors
3 and 5
1 and 15
Difference of
Two Squares
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Nat 5
Now try N5 TJ
Ex 7.3 Q1 & Q3
Ch7 (page 68)
11-May-17
Created by Mr. [email protected]
Starter Questions
Nat 5
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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
11-May-17
3ab – b2
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Summary of
Factorising
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Nat 5
Learning Intention
1. To explain the factorising
priorities.
11-May-17
Success Criteria
1. Be able use the factorise
priorities to factorise
various expressions.
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Summary of
Factorising
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Nat 5
When we are asked to factorise there is priority we
must do it in.
1.
Take any common factors out and put
Onlythem
TWO
outside the brackets.
terms
2.
Check for the difference of two squares.
3.
Factorise any quadratic expression left.
THREE terms
11-May-17
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If you can
successfully
Difference
of
complete this exercise
then you have the
Two Squares
necessary skills to pass
Nat 5
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the algebraic part of the
course.
Now try N5 TJ
Ex 7.3 Q4
Ch7 (page 68)
11-May-17
Created by Mr. [email protected]