Download S3_Credit_Factorisin..

Algebraic Operations
S3
Credit
www.mathsrevision.com
Factors / HCF
Common Factors
Difference of Squares
Factorising Trinomials (Quadratics)
Factor Priority
5-May-17
Created by Mr. [email protected]
Starter Questions
S3
Credit
www.mathsrevision.com
Q1.
Multiply out
(a)
a (4y – 3x)
(b)
(2x-1)(x+4)
Q2. True or false
5 1 4
1
  1
6 3 3
3
Q3. Write down all the number that divide
into 12 without leaving a remainder.
5-May-17
Created by Mr. [email protected]
Factors
www.mathsrevision.com
S3
Credit
Using Factors
Learning Intention
1. To explain that a factor
divides into a number
without leaving a
remainder
2. To explain how to find
Highest Common Factors
5-May-17
Success Criteria
1. To identify factors using
factor pairs
2. Find HCF for two numbers
by comparing factors.
Created by Mr. [email protected]
Factors
www.mathsrevision.com
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
4 and 14
7 and
5-May-17
8
Created by Mr. [email protected]
Factors
www.mathsrevision.com
S3
Credit
Highest Common Factor
Highest
Common
Factor
Largest
Same
Number
We need to write out all factor pairs in order
to find
the Highest Common Factor.
5-May-17
Created by Mr. [email protected]
Factors
www.mathsrevision.com
S3
Credit
Highest Common Factor
Example :
Find the HCF of 8 and 12.
F8 = 1 and 8
F12 = 1 and 12
2 and 4
2 and 6
3 and 4
HCF = 4
5-May-17
Created by Mr. [email protected]
Factors
www.mathsrevision.com
S3
Credit
Highest Common Factor
Example :
Find the HCF of 4x and x2.
F4x = 1, and 4x
2 and 2x
4 and
Example :
Fx2 = 1 and x2
x and x
x
HCF = x
Find the HCF of 5 and 10x.
F5 = 1 and 5
5-May-17
F10x = 1, and 10x
2 and 5x
HCF = 5
5 and 2x
Created by Mr. [email protected]
10 and x
Factors
www.mathsrevision.com
S3
Credit
Highest Common Factor
Example :
Find the HCF of ab and 2b.
F ab = 1 and ab
F2b = 1 and 2b
a and b
Example :
Find the HCF of 2h2 and 4h.
F 2h2 = 1 and 2h2
2 and h2 , h and 2h
5-May-17
2 and
F4h = 1 and 4h
HCF = 2h
Created by Mr. [email protected]
2 and 2h
4 and h
b
HCF = b
Factors
S3
Credit
www.mathsrevision.com
Find the HCF for these terms
(a)
16w and 24w
8w
(b)
9y2 and 6y
3y
(c)
4h and 12h2
4h
(d)
ab2 and a2b
ab
5-May-17
Created by Mr. [email protected]
Factors
www.mathsrevision.com
S3
Credit
Now try Ex 2.1 & 3.1
First Column in each
Question
Ch5 (page 86)
5-May-17
Created by Mr. Lafferty
Starter Questions
www.mathsrevision.com
S3
Credit
Q1.
Expand out
(a)
a (4y – 3x) -2ay
(b)
(x + 5)(x - 5)
Q2. Write out in full
4.85 10
3
Q3. True or False all the factors of 5x2 are
1, x, 5
5-May-17
Created by Mr. [email protected]
Factorising
www.mathsrevision.com
S3
Credit
Using Factors
Learning Intention
1. To show how to factorise
terms using the Highest
Common Factor and one
bracket term.
5-May-17
Success Criteria
1. To identify the HCF for
given terms.
2. Factorise terms using the
HCF and one bracket term.
Created by Mr. [email protected]
Check by multiplying
Factorising
S3
Credit
www.mathsrevision.com
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
5-May-17
15 ÷ 3 = 5
Created by Mr. [email protected]
3
3(
)
3( x + 5 )
Check by multiplying
Factorising
S3
Credit
www.mathsrevision.com
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
5-May-17
6xy ÷ 2x = 3y
Created by Mr. [email protected]
2x
2x(
)
2x( 2x- 3y )
Factorising
S3
Credit
www.mathsrevision.com
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)
5-May-17
Created by Mr. [email protected]
Factorising
www.mathsrevision.com
S3
Credit
Now try Ex 4.1 & 4.2
First 2 Columns only
Ch5 (page 88)
5-May-17
Created by Mr. [email protected]
Starter Questions
S3
Credit
www.mathsrevision.com
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
Q3. Write down the arithmetic operation
associated with the word ‘difference’.
5-May-17
Created by Mr. [email protected]
Difference of
Two Squares
www.mathsrevision.com
S3
Credit
Learning Intention
1. To show 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.
5-May-17
Created by Mr. [email protected]
Difference of
Two Squares
www.mathsrevision.com
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
a2 – b2
First
square term
5-May-17
Difference
Second
square term
Created by Mr. [email protected]
Difference of
Two Squares
www.mathsrevision.com
S3
Credit
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 5-May-17
Created by Mr. [email protected]
Difference of
Two Squares
www.mathsrevision.com
S3
Credit
Keypoints
Format
a2 – b2
Always the difference sign ( a + b )( a – b )
5-May-17
Created by Mr. Lafferty
Difference of
Two Squares
S3
Credit
www.mathsrevision.com
Factorise using the difference of two squares
(a)
x2 – 72
(x + 7 )( x – 7 )
(b)
w2 – 1
( w + 1 )( w – 1 )
(c)
9a2 – b2
(d)
16y2 – 100k2
5-May-17
( 3a + b )( 3a – b )
( 4y + 10k )( 4y – 10k )
Created by Mr. Lafferty
Difference of
Two Squares
www.mathsrevision.com
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
First take out common factor
2(a2 - 9)
Now apply the difference of two squares
2( a + 3 )( a – 3 )
5-May-17
Created by Mr. Lafferty
Difference of
Two Squares
S3
Credit
www.mathsrevision.com
Factorise these trickier expressions.
(a)
6x2 – 24
6(x + 2 )( x – 2 )
(b)
3w2 – 3
3( w + 1 )( w – 1 )
(c)
8 – 2b2
(d)
5-May-17
27w2 – 12
2( 2 + b )( 2 – b )
3(3 w + 2 )( 3w – 2 )
Created by Mr. Lafferty
Difference of
Two Squares
www.mathsrevision.com
S3
Credit
Now try Ex 5.1 & 5.2
First 2 Columns only
Ch5 (page 90)
5-May-17
Created by Mr. [email protected]
Starter Questions
S3
Credit
www.mathsrevision.com
Q1.
True or false
y ( y + 6 ) -7y = y2 -7y + 6
Q2. Fill in the ?
49 – 4x2 = ( ? + ?x)(? – 2?)
Q3. Write in scientific notation 0.0341
5-May-17
Created by Mr. [email protected]
Factorising
Using St. Andrew’s Cross method
www.mathsrevision.com
S3
Credit
Learning Intention
1. To show how to factorise
trinomials ( quadratics)
using
St. Andrew's Cross method.
5-May-17
Success Criteria
1. Understand the steps of the
St. Andrew’s Cross method.
2. Be able to factorise
quadratics using SAC method.
Created by Mr. [email protected]
Factorising
Using St. Andrew’s Cross method
www.mathsrevision.com
S3
Credit
There are 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.
5-May-17
Created by Mr. [email protected]
Removing
Double Brackets
www.mathsrevision.com
S3
Credit
A LITTLE REVISION
Multiply out the brackets and Simplify
(x + 1)(x + 2)
1.
Write down
2.
Tidy up !
5-May-17
F
O
I
L
x2 + 2x + x + 2
x2 + 3x + 2
Created by Mr. [email protected]
Factorising
Using St. Andrew’s Cross method
S3
Credit
www.mathsrevision.com
We use the SAC method to go the opposite way
FOIL
(x + 1)(x + 2)
(x + 1)(x + 2)
5-May-17
SAC
Created by Mr. [email protected]
x2 + 3x+ 2
x2 + 3x+ 2
Factorising
Using St. Andrew’s Cross method
S3
Credit
www.mathsrevision.com
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
(
)(
5-May-17
)
Created by Mr. [email protected]
(+2) x( +1) = +2
(+2x) +( +1x) = +3x
Factorising
Using St. Andrew’s Cross method
S3
Credit
www.mathsrevision.com
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
(
5-May-17
)(
)
Created by Mr. [email protected]
(+5) x( +1) = +5
(+5x) +( +1x) = +6x
Factorising
One number
Using St. Andrew’s Cross method
must be +
and one -
S3
Credit
www.mathsrevision.com
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
(
5-May-17
)(
)
Created by Mr. [email protected]
(+4) x( -3) = -12
(+4x) +( -3x) = +x
Factorising
Both numbers
Using St. Andrew’s Cross method
must be -
S3
Credit
www.mathsrevision.com
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
(
5-May-17
)(
)
Created by Mr. [email protected]
(-2) x( -2) = +4
(-2x) +( -2x) = -4x
Factorising
One number
Using St. Andrew’s Cross method
must be +
and one -
S3
Credit
www.mathsrevision.com
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
(
5-May-17
)(
)
Created by Mr. [email protected]
(-3) x( +1) = -3
(-3x) +( x) = -2x
Factorising
Using St. Andrew’s Cross method
S3
Credit
www.mathsrevision.com
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 )
5-May-17
Created by Mr. Lafferty
Factorising
Using St. Andrew’s Cross method
www.mathsrevision.com
S3
Credit
Now try Ex6.1
Ch5 (page 93)
5-May-17
Created by Mr. [email protected]
Starter Questions
www.mathsrevision.com
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.
Q2. Factorise
5-May-17
2 + x – x2
Created by Mr. [email protected]
Factorising
Using St. Andrew’s Cross method
www.mathsrevision.com
S3
Credit
Learning Intention
1. To show how to factorise
trinomials ( quadratics) of
the form ax2 + bx +c using
SAC.
5-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 -
S3
Credit
www.mathsrevision.com
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
(
5-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 -
S3
Credit
www.mathsrevision.com
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
(
5-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 -
www.mathsrevision.com
S3
Credit
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
(
5-May-17
)(
)
Can we do it !
Created by Mr. [email protected]
Factorising
Using St. Andrew’s Cross method
www.mathsrevision.com
S3
Credit
Find another set of
factors for LHS
4x2 - 4x - 3
2x
-3
2x
+1
(
5-May-17
)(
Repeat the factors for
RHS to see if it
factorises now
)
Created by Mr. [email protected]
Factors
1 and -3
-1 and 3
Factorising
Both numbers
Using St. Andrew’s Cross method
must be +
www.mathsrevision.com
S3
Credit
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
(
5-May-17
)(
)
Can we do it !
Created by Mr. [email protected]
Factorising
Using St. Andrew’s Cross method
www.mathsrevision.com
S3
Credit
Find another set of
factors for LHS
8x2+22x+15
4x
+5
2x
+3
(
5-May-17
)(
Repeat the factors for
RHS to see if it
factorises now
)
Created by Mr. [email protected]
Factors
3 and 5
1 and 15
Factorising
Using St. Andrew’s Cross method
www.mathsrevision.com
S3
Credit
Now try Ex 7.1
First 2 columns only
Ch5 (page 95)
5-May-17
Created by Mr. [email protected]
Starter Questions
www.mathsrevision.com
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.
Q3. True or false
5-May-17
3a2 b – ab2 =a2b2(3b – a)
Created by Mr. [email protected]
Summary of
Factorising
www.mathsrevision.com
S3
Credit
Learning Intention
1. To explain the factorising
priorities.
5-May-17
Success Criteria
1. Be able use the factorise
priorities to factorise
various expressions.
Created by Mr. [email protected]
Summary of
Factorising
www.mathsrevision.com
S3
Credit
When we are asked to factorise there is priority we
must do it in.
1.
Take any common factors out and put them
outside the brackets.
2.
Check for the difference of two squares.
3.
5-May-17
Factorise any quadratic expression left.
Created by Mr. [email protected]
Summary of
Factorising
www.mathsrevision.com
S3
Credit
St. Andrew’s Cross method
2
Difference
squares
Take Out Common Factor
5-May-17
Created by Mr. [email protected]
If youof
can successfully
Summary
complete this exercise
then you have the
Factorising
necessary skills to pass
www.mathsrevision.com
S3
Credit
the algebraic part of the
course.
Now try Ex 8.1
Ch5 (page 97)
5-May-17
Created by Mr. [email protected]