Algebraic Operations

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

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)

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

Int 2

3(b + 5) =3b + 15

Example 1

4(w - 2) =4w - 8

Example 2

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Int 2

a(y - 1) =ay - a

Example 3

p(w - 6) =pw- 6p

Example 4

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Int 2

x(x + 3) =x2 + 3x

Example 5

3q(3q -2m) =9q2 - 6mq

Example 6

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Int 2

-2(h + 5) =-2h - 10

Example 7

-(g - 9) = -g + 9

Example 8

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Int 2

6(x + 4) =6x+ 24

Example 9

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Find my Area

(x + 4)

6

Int 2

8 +2(h + 3) =8 + 6

Example 10

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+ 2hNow tidy up !

+ 14= 2h

Int 2

-2(y - 1) + 4 =-2y + 4

Example 10

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+ 2Now tidy up !

+ 6= -2y

Int 2

y - (4 - y) = y + y

Example 11

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- 4

= - 4Now tidy up !

Int 2

x(x + 6)

Example 12

Find the area ofthe picture frame.

Apr 22, 2023Apr 22, 2023


(x + 6)

4x(x + 4)

Area = 4(x + 4) –

Int 2

x(x + 6) – 4(x + 4)

x2 + 6x

Example 12

Apr 22, 2023Apr 22, 2023


Area =

- 4x - 16

x2 + 2x - 16Now

tidy up !

Int 2

x(x - 3) + 2(x - 3)

x2 - 3x

Example 13

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+ 2x - 6

x2 - x - 6Now

tidy up !

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Now try Exercise 1

Ch5 MIA (page 48)

Int 2


Apr 22, 2023Apr 22, 2023


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

Apr 22, 2023Apr 22, 2023


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

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

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 !


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


1. Write down F O I L- x - 2

2. Tidy up !

Int 2

Apr 22, 2023Apr 22, 2023

(x + 1)(x - 2)


(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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Now try Exercise 2Q1

Ch5 MIA (page 50)

Int 2


Int 2

“the wee table method”

Apr 22, 2023Apr 22, 2023


(y + 2)(y + 5) y+ 2 y+ 5

We haveMultiplicatio

n Table

+5y

+10+2y

y2 Tidy up !

y2 + 7y +10

Int 2

Example 2

Apr 22, 2023Apr 22, 2023


(2x - 1)(x + 3) 2x - 1 x+ 3

Be careful with the negative

signs

+6x

-3-x

2x2 Tidy up !

2x2 + 5x - 3

Int 2

Example 3

Apr 22, 2023Apr 22, 2023


(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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Now try Exercise 2

Ch5 MIA (page 50)

Int 2


Apr 22, 2023Apr 22, 2023


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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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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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

F8 = 1 and 8

2 and 4

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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

F4x =1, and 4x ,

2 and 2x

4 and x

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Example : Find the HCF of 4x and x2.

HCF = x

Fx2 = 1 and x2

x and x


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

F ab = 1 and ab

a and b

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Example : Find the HCF of ab and 2b.

HCF = b

Fx2 = 1 and 2b

2 and b


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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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

22 Apr 202322 Apr 2023

Now try Exercise 3Q3 and Q4

Ch5 (page 52)

FactorsFactorsInt 2

Apr 22, 2023Apr 22, 2023



(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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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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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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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


you started

Int 2

22 Apr 202322 Apr 2023


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

22 Apr 202322 Apr 2023

Now try Exercise 4Start at Q2

Ch5 (page 53)

FactorisingFactorisingInt 2

Apr 22, 2023Apr 22, 2023



(a) a (8 – 3x + 6a) =

Q2. Factorise 3x2 – 6x

Int 2

Q3. Write down the first 10 square numbers.

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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


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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 -


you started


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Keypoints

Format a2 – b2

Always the difference sign -

( a + b )( a – b )


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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 )


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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 )


First take out common factor 2(a2 - 9)

Now apply the difference of two squares

22 Apr 202322 Apr 2023

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 )


22 Apr 202322 Apr 2023

Now try Exercise 5

Ch5 (page 54)


Apr 22, 2023Apr 22, 2023


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

22 Apr 202322 Apr 2023


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

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.

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


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


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


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


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

?

22 Apr 202322 Apr 2023


(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


22 Apr 202322 Apr 2023

Now try Exercise 6

Ch5 (page 56)


Apr 22, 2023Apr 22, 2023


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.

22 Apr 202322 Apr 2023


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


Int 2


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

Int 2

Harder Still

+ 22x

( )( )

Apr 22, 2023Apr 22, 2023

Put down two brackets

8x2 +22 x + 15 8x2

+15

F O+I L


8x x x = 8x2

or4x x 2x = 8x2

15 x 1 = 15or

3 x 5 = 15

Int 2

We just have to try all combinations to see what works.

Apr 22, 2023Apr 22, 2023

+121x


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

22 Apr 202322 Apr 2023


(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


22 Apr 202322 Apr 2023

Now try Exercise 7

Ch5 (page 57)


Apr 22, 2023Apr 22, 2023


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

22 Apr 202322 Apr 2023


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

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.

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.

Algebraic Operations

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x2 x

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asingle bracketint

int 2q3

asingle bracketx