algebraic operations
DESCRIPTION
Algebraic Operations. Removing Brackets. Pairs of Brackets. Factors. Common Factors. Difference of Squares. Factorising Trinomials (Quadratics). Factor Priority. Int 2. Q1.Calculate (a)-3 x 5 =(b)-6 x -7 =. Starter Questions. Q2.Calculate (a)w x w =(b)-2a x 4a =. - PowerPoint PPT PresentationTRANSCRIPT
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Int 2
Algebraic Algebraic OperationsOperations
Removing Brackets
Difference of Squares
Pairs of Brackets
Factors
Common Factors
Factorising Trinomials (Quadratics)
Factor Priority
Apr 22, 2023Apr 22, 2023
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Apr 22, 2023Apr 22, 2023
Starter QuestionsStarter Questions
Q1. Calculate
(a) -3 x 5 = (b) -6 x -7 =
Q2. Calculate
(a) w x w = (b) -2a x 4a =
Int 2
Q3. Find the gradient of the line if (3, 7) and(12, 34)
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Apr 22, 2023Apr 22, 2023
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1. To show how to multiply out (remove) a single bracket.
1.1. Understand the keypoints Understand the keypoints of multiplying out a of multiplying out a expression with a single expression with a single bracket.bracket.
Int 2
2.2. Be able multiply out a Be able multiply out a expression with a single expression with a single bracket.bracket.
Removing aRemoving aSingle BracketSingle Bracket
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Int 2
3(b + 5) =3b + 15
Example 1
4(w - 2) =4w - 8
Example 2
Apr 22, 2023Apr 22, 2023
Removing aRemoving aSingle BracketSingle Bracket
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Int 2
a(y - 1) =ay - a
Example 3
p(w - 6) =pw- 6p
Example 4
Apr 22, 2023Apr 22, 2023
Removing aRemoving aSingle BracketSingle Bracket
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Int 2
x(x + 3) =x2 + 3x
Example 5
3q(3q -2m) =9q2 - 6mq
Example 6
Apr 22, 2023Apr 22, 2023
Removing aRemoving aSingle BracketSingle Bracket
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Int 2
-2(h + 5) =-2h - 10
Example 7
-(g - 9) = -g + 9
Example 8
Apr 22, 2023Apr 22, 2023
Removing aRemoving aSingle BracketSingle Bracket
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Int 2
6(x + 4) =6x+ 24
Example 9
Apr 22, 2023Apr 22, 2023
Removing aRemoving aSingle BracketSingle Bracket
Find my Area
(x + 4)
6
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Int 2
8 +2(h + 3) =8 + 6
Example 10
Apr 22, 2023Apr 22, 2023
Removing aRemoving aSingle BracketSingle Bracket
+ 2hNow tidy up !
+ 14= 2h
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Int 2
-2(y - 1) + 4 =-2y + 4
Example 10
Apr 22, 2023Apr 22, 2023
Removing aRemoving aSingle BracketSingle Bracket
+ 2Now tidy up !
+ 6= -2y
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Int 2
y - (4 - y) = y + y
Example 11
Apr 22, 2023Apr 22, 2023
Removing aRemoving aSingle BracketSingle Bracket
- 4
= - 4Now tidy up !
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Int 2
x(x + 6)
Example 12
Find the area ofthe picture frame.
Apr 22, 2023Apr 22, 2023
Removing aRemoving aSingle BracketSingle Bracket
(x + 6)
4x(x + 4)
Area = 4(x + 4) –
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Int 2
x(x + 6) – 4(x + 4)
x2 + 6x
Example 12
Apr 22, 2023Apr 22, 2023
Removing aRemoving aSingle BracketSingle Bracket
Area =
- 4x - 16
x2 + 2x - 16Now
tidy up !
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Int 2
x(x - 3) + 2(x - 3)
x2 - 3x
Example 13
Apr 22, 2023Apr 22, 2023
Removing aRemoving aSingle BracketSingle Bracket
+ 2x - 6
x2 - x - 6Now
tidy up !
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22 Apr 202322 Apr 2023
Now try Exercise 1
Ch5 MIA (page 48)
Int 2
Removing aRemoving aSingle BracketSingle Bracket
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Apr 22, 2023Apr 22, 2023
Starter QuestionsStarter Questions
Q1. Calculate
(a) -3y x 5y =(b) -6q x (-4q) =
Q2. Calculate
(a) a(b - c) = (b) -2a( b – a) =
Int 2
Q3. Write down the gradient and were the line cuts the y – axis. y = 5 – 3x
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Apr 22, 2023Apr 22, 2023
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1. To show 2 methods for multiplying out brackets
1.1. Understand the keypoints Understand the keypoints of multiplying out double of multiplying out double brackets.brackets.
Int 2
2.2. Be able multiply out Be able multiply out double brackets using 2 double brackets using 2 methods.methods.
Removing Removing Double BracketsDouble Brackets
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Int 2
There two methods we can use to multiply out DOUBLE brackets.
Apr 22, 2023Apr 22, 2023
RemovingRemovingDouble BracketsDouble Brackets
Simply remember the word
F O I LMultiply First 2
Multiply Last 2
Multiply Outside 2
Multiply Inside 2
First Method
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Int 2
(x + 1)(x + 2)
x2 + 2x
Example 1 : Multiply out the brackets and Simplify
Apr 22, 2023Apr 22, 2023 Created by Mr. Created by Mr. [email protected]@mathsrevision.com
1. Write down F O I L+ x + 2
2. Tidy up !
RemovingRemovingDouble BracketsDouble Brackets
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Int 2
(x - 1)(x + 2)
x2 + 2x
Example 2 : Multiply out the brackets and Simplify
Apr 22, 2023Apr 22, 2023 Created by Mr. Created by Mr. [email protected]@mathsrevision.com
Removing aRemoving aSingle BracketSingle Bracket
1. Write down F O I L- x - 2
2. Tidy up !
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Int 2
Apr 22, 2023Apr 22, 2023
(x + 1)(x - 2)
RemovingRemovingDouble BracketsDouble Brackets
(x - 1)(x - 2)
(x + 3)(x + 2)
(x - 3)(x + 2)
(x + 3)(x - 2)
x2 - x - 2
x2 - 3x + 2
x2 + 5x + 6
x2 - x - 6
x2 + x - 6
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22 Apr 202322 Apr 2023
Now try Exercise 2Q1
Ch5 MIA (page 50)
Int 2
Removing aRemoving aSingle BracketSingle Bracket
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Int 2
“the wee table method”
Apr 22, 2023Apr 22, 2023
RemovingRemovingDouble BracketsDouble Brackets
(y + 2)(y + 5) y+ 2 y+ 5
We haveMultiplicatio
n Table
+5y
+10+2y
y2 Tidy up !
y2 + 7y +10
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Int 2
Example 2
Apr 22, 2023Apr 22, 2023
RemovingRemovingDouble BracketsDouble Brackets
(2x - 1)(x + 3) 2x - 1 x+ 3
Be careful with the negative
signs
+6x
-3-x
2x2 Tidy up !
2x2 + 5x - 3
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Int 2
Example 3
Apr 22, 2023Apr 22, 2023
RemovingRemovingDouble BracketsDouble Brackets
(x + 4)(x2 + 3x + 2) x+ 4 x2 + 3x
Just a biggerMultiplicatio
n Table
+3x2
+12x+4x2
x3 Tidy up !
x3 + 7x2 + 14x + 8
+ 2
+2x
+8
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22 Apr 202322 Apr 2023
Now try Exercise 2
Ch5 MIA (page 50)
Int 2
Removing aRemoving aSingle BracketSingle Bracket
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Apr 22, 2023Apr 22, 2023
Starter QuestionsStarter Questions
Q1. Remove the brackets
(a) a (4y – 3x) = (b) (2x-1)(x+4) =
Q2. Calculate The interest on £20 over 5 years @ a compound interest of 7% per year.
Int 2
Q3. Write down all the number that divide into 12 without leaving a remainder.
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22 Apr 202322 Apr 2023
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1.1. To identify factors using To identify factors using factor pairsfactor pairs
1. To explain that a factor divides into a number without leaving a remainder
2. To explain how to find Highest Common Factors
2.2. Find HCF for two numbers Find HCF for two numbers by comparing factors.by comparing factors.
FactorsFactorsUsing FactorsInt 2
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22 Apr 202322 Apr 2023
FactorsFactorsFactors
Example : Find the factors of 56.
F56 = 1 and 56
Always divide by 1 and find its pair
2 and 28
4 and 14
7 and 8
From 2 find other factors and their pairs
Int 2
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FactorsFactorsHighest Common Factor
We need to write out all factor pairs in order to find
the Highest Common Factor.
HighestCommonFactor
LargestSameNumber
Int 2
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F8 = 1 and 8
2 and 4
22 Apr 202322 Apr 2023
Example : Find the HCF of 8 and 12.
HCF = 4
F12 = 1 and 12
2 and 6
3 and 4
Highest Common Factor
FactorsFactorsInt 2
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F4x =1, and 4x ,
2 and 2x
4 and x
22 Apr 202322 Apr 2023
Example : Find the HCF of 4x and x2.
HCF = x
Fx2 = 1 and x2
x and x
Highest Common Factor
F5 = 1 and 5
Example : Find the HCF of 5 and 10x.
HCF = 5
F10x = 1, and 10x
2 and 5x , 5 and 2x
10 and x
FactorsFactorsInt 2
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F ab = 1 and ab
a and b
22 Apr 202322 Apr 2023
Example : Find the HCF of ab and 2b.
HCF = b
Fx2 = 1 and 2b
2 and b
Highest Common Factor
F 2h2 = 1 and 2h2
2 and h2 , h and 2h
Example : Find the HCF of 2h2 and 4h.
HCF = 2h
F4h = 1 and 4h
2 and 2h
4 and h
FactorsFactorsInt 2
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22 Apr 202322 Apr 2023
FactorsFactors
Find the HCF for these terms
(a) 16w and 24w
(b) 9y2 and 6y
(c) 4h and 12h2
(d) ab2 and a2b
8w
3y
4h
ab
Int 2
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22 Apr 202322 Apr 2023
Now try Exercise 3Q3 and Q4
Ch5 (page 52)
FactorsFactorsInt 2
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Apr 22, 2023Apr 22, 2023
Starter QuestionsStarter Questions
Q1. Remove the brackets
(a) a (4y – 3x) = (b) (x + 5)(x - 5) =
Q2. For the line y = -x + 5, find the gradientand where it cuts the y axis.
Int 2
Q3. Find the highest common factor forp2q and pq2.
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22 Apr 202322 Apr 2023
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1.1. To identify the HCF for To identify the HCF for given terms.given terms.
1. To show how to factorise terms using the Highest Common Factor and one bracket term.
2.2. Factorise terms using the Factorise terms using the HCF and one bracket term.HCF and one bracket term.
FactorisingFactorisingUsing FactorsInt 2
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22 Apr 202322 Apr 2023
FactorisingFactorising
Example Factorise 3x + 15
1. Find the HCF for 3x and 15 3
2. HCF goes outside the bracket 3( )
3. To see what goes inside the bracketdivide each term by HCF
3x ÷ 3 = x 15 ÷ 3 = 5 3( x + 5 )
Check by multiplying out the bracket to get back to where
you started
Int 2
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22 Apr 202322 Apr 2023
FactorisingFactorising
Example
1. Find the HCF for 4x2 and 6xy 2x
2. HCF goes outside the bracket 2x( )
3. To see what goes inside the bracketdivide each term by HCF
4x2 ÷ 2x =2x 6xy ÷ 2x = 3y 2x( 2x- 3y )
Factorise 4x2 – 6xy
Check by multiplying out the bracket to get back to where
you started
Int 2
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22 Apr 202322 Apr 2023
FactorisingFactorising
Factorise the following :
(a) 3x + 6
(b) 4xy – 2x
(c) 6a + 7a2
(d) y2 - y
3(x + 2)
2x(y – 1)
a(6 + 7a)
y(y – 1)
Be careful !
Int 2
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22 Apr 202322 Apr 2023
Now try Exercise 4Start at Q2
Ch5 (page 53)
FactorisingFactorisingInt 2
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Apr 22, 2023Apr 22, 2023
Starter QuestionsStarter Questions
Q1. Remove the brackets
(a) a (8 – 3x + 6a) =
Q2. Factorise 3x2 – 6x
Int 2
Q3. Write down the first 10 square numbers.
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22 Apr 202322 Apr 2023
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1.1. Recognise when we Recognise when we have a difference of have a difference of two squares.two squares.
1. To show how to factorise the special case of the difference of two squares.
2.2. Factorise the difference of Factorise the difference of two squares.two squares.
Difference of Difference of Two SquaresTwo SquaresInt 2
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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
a2 – b2
First square term
Secondsquare term
Difference
Difference of Difference of Two SquaresTwo SquaresInt 2
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a2 – b2
First square term
Secondsquare term
Difference
This factorises to
( a + b )( a – b )
Two brackets the same except for + and a -
Check by multiplying out the bracket to get back to where
you started
Difference of Difference of Two SquaresTwo SquaresInt 2
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Keypoints
Format a2 – b2
Always the difference sign -
( a + b )( a – b )
Difference of Difference of Two SquaresTwo SquaresInt 2
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Factorise using the difference of two squares
(a) x2 – y2
(b) w2 – z2
(c) 9a2 – b2
(d) 16y2 – 100k2
(x + y )( x – y )
( w + z )( w – z )
( 3a + b )( 3a – b )
( 4y + 10k )( 4y – 10k )
Difference of Difference of Two SquaresTwo SquaresInt 2
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Trickier type of questions to factorise.Sometimes we need to take out a commonAnd the use the difference of two squares.
ExampleFactorise 2a2 - 18
2( a + 3 )( a – 3 )
Difference of Difference of Two SquaresTwo SquaresInt 2
First take out common factor 2(a2 - 9)
Now apply the difference of two squares
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Factorise these trickier expressions.
(a) 6x2 – 24
(b) 3w2 – 3
(c) 8 – 2b2
(d) 27w2 – 12
6(x + 2 )( x – 2 )
3( w + 1 )( w – 1 )
2( 2 + b )( 2 – b )
3(3 w + 2 )( 3w – 2 )
Difference of Difference of Two SquaresTwo SquaresInt 2
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22 Apr 202322 Apr 2023
Now try Exercise 5
Ch5 (page 54)
Difference of Difference of Two SquaresTwo SquaresInt 2
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Apr 22, 2023Apr 22, 2023
Starter QuestionsStarter Questions
Q1. Multiple out the brackets and simplify.
(a) ( y – 3 )( y + 6 )
Q2. Factorise 49 – 4x2
Int 2
Q3. Write down an equation parallel to y = 4x + 1
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22 Apr 202322 Apr 2023
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1.1. Be able to factorise Be able to factorise quadratics using FOIL.quadratics using FOIL.
1. To show how to factorise trinomials ( quadratics) using FOIL.
Int 2
FactorisingFactorisingUsing FOILUsing FOIL
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22 Apr 202322 Apr 2023
FactorisingFactorisingUsing FOILUsing FOILInt 2
There various ways of factorising trinomials ( quadratics)e.g. The ABC method, St. Andrew’s cross method.
We will use our previous knowledge and use the
FOIL METHOD to factorise quadratics.
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Int 2
(x + 1)(x + 2)
x2 + 2x
A LITTLE REVISIONMultiply out the brackets and Simplify
Apr 22, 2023Apr 22, 2023
1. Write down F O I L+ x + 2
2. Tidy up !x2 + 3x + 2
RemovingRemovingDouble BracketsDouble Brackets
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Int 2
(x + 1)(x + 2) x2 + 3x
We can also use FOIL to go the opposite way
Apr 22, 2023Apr 22, 2023
+ 2FOIL
(x + 1)(x + 2) x2 + 3x + 2FOIL
FactorisingFactorisingUsing FOILUsing FOIL
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Int 2
+ 3x( )( )
+ 2x
Apr 22, 2023Apr 22, 2023
Put down two brackets + x
x2 + 3x+2Strategy for factorising quadratics
x2
+2
x x x =
1 x 2 =x x+1 + 2
F O+I L
FactorisingFactorisingUsing FOILUsing FOIL
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Int 2
Sometimes it can be trick to get O+I correct
+ x( )( )
+ 4x
Apr 22, 2023Apr 22, 2023
Put down two brackets -3 x
x2 + x - 12 x2
-12
x x x =
-3 x 4 =
x x-3 + 4
F O+I L
FactorisingFactorisingUsing FOILUsing FOIL
O+I value will be(-1)x + 12x = 11x1x + (-12x) = -11x
(-2x) + 6x = 4x2x + (-6x) = -4x(-3x) + 4x = +x
3x + (-4) = -x
(-1) x 12 = -121 x (-12) = -12(-2) x 6 = -122 x (-6) = -12(-3) x 4 = -123 x ( 4) = -12
?
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22 Apr 202322 Apr 2023
Factorise using the difference of two squares
(a) m2 + 2m +1
(b) y2 + 6m + 5
(c) b2 – b -2
(d) a2 – 5a + 6
(m + 1 )( m + 1 )
( y + 5 )( y + 1 )
( b - 2 )( b + 1 )
( a - 3 )( a – 2 )
Int 2
FactorisingFactorisingUsing FOILUsing FOIL
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22 Apr 202322 Apr 2023
Now try Exercise 6
Ch5 (page 56)
FactorisingFactorisingUsing FOILUsing FOILInt 2
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Apr 22, 2023Apr 22, 2023
Starter QuestionsStarter Questions
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 50cmand diameter 10cm
Int 2
Q3. A line has gradient -7 and cuts the y axisat -5. Write down the equation of the line.
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22 Apr 202322 Apr 2023
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1.1. Be able to factorise Be able to factorise quadratics using FOIL.quadratics using FOIL.
1. To show how to factorise trinomials ( quadratics) using FOIL.
Int 2
FactorisingFactorisingUsing FOILUsing FOIL
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Int 2
FactorisingFactorisingUsing FOILUsing FOIL
3x x x =3x2
( )( ) - 4
(-1) x 4 = -41 x (-4) = -4(-4) x 1 = -44 x (-1) = -4
?Slightly harder example
- x
+ 3x
Apr 22, 2023Apr 22, 2023
Put down two brackets - 4 x
3x2 - x - 4
-4-4 x 1 =
3x x+ 1
F O+I L
O+I value will be12x + (-1x) = 11x(-12x) + 1x = -11x
3x - 4x = -x-3x + 4x = x
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Int 2
Harder Still
+ 22x
( )( )
Apr 22, 2023Apr 22, 2023
Put down two brackets
8x2 +22 x + 15 8x2
+15
F O+I L
FactorisingFactorisingUsing FOILUsing FOIL
8x x x = 8x2
or4x x 2x = 8x2
15 x 1 = 15or
3 x 5 = 15
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Int 2
We just have to try all combinations to see what works.
Apr 22, 2023Apr 22, 2023
+121x
FactorisingFactorisingUsing FOILUsing FOIL
8x x x = 8x2
(8x + 1)(x+15)
(8x + 15)(x+1)
(8x + 3)(x+ 5)
(8x + 5)(x+ 3)
4x x 2x = 8x2
(4x + 1)(2x+15)
(4x + 15)(2x+1)
(4x + 3)(2x+ 5)
(4x + 5)(2x+ 3)
+23x
+29x
+43x
+62x
+34x
+26x
+22x
Middle termO+I = +22x
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22 Apr 202322 Apr 2023
Factorise using the difference of two squares
(a) m2 + 2m +1
(b) y2 + 6m + 5
(c) b2 – b - 2
(d) a2 – 5a + 6
(m + 1 )( m + 1 )
( y + 5 )( y + 1 )
( b - 2 )( b + 1 )
( a - 3 )( a – 2 )
Int 2
FactorisingFactorisingUsing FOILUsing FOIL
![Page 66: Algebraic Operations](https://reader035.vdocuments.site/reader035/viewer/2022070417/56815569550346895dc33358/html5/thumbnails/66.jpg)
22 Apr 202322 Apr 2023
Now try Exercise 7
Ch5 (page 57)
FactorisingFactorisingUsing FOILUsing FOILInt 2
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Apr 22, 2023Apr 22, 2023
Starter QuestionsStarter Questions
Q1. Multiple out the brackets and simplify.
(a) ( 2x – 5 )( x + 5 )
Int 2
Q3. Find the gradient and where line cut y-axis.x = y + 1
Q2. Find the volume of a cylinder with high 6mand diameter 9cm
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22 Apr 202322 Apr 2023
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1.1. Be able use the factorise Be able use the factorise priorities to factorise priorities to factorise various expressions.various expressions.
1. To explain the factorising priorities.
Int 2
Summary ofSummary ofFactorisingFactorising
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22 Apr 202322 Apr 2023
Summary ofSummary ofFactorisingFactorisingInt 2
When we are asked to factorise there is priority wemust do it in.
1. 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.
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22 Apr 202322 Apr 2023
Now try Exercise 8
Ch5 (page 57)
Int 2
Summary ofSummary ofFactorisingFactorising
If you can successfully complete this exercise
then you have the necessary skills to pass the algebraic part of the
course.