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Algebraic OperationsAlgebraic Operations

Apr 21, 2023Apr 21, 2023 Created by Mr. Created by Mr. Lafferty@mathsrevision.comLafferty@mathsrevision.com

Difference of Squares

Factors / HCF

Common Factors

Factorising Trinomials (Quadratics)Factor Priority

See Quadratic Theory for Exam Questions

Starter QuestionsStarter Questions

Apr 21, 2023Apr 21, 2023 Created by Mr. Created by Mr. Lafferty@mathsrevision.comLafferty@mathsrevision.comw

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.com Q1. Remove the brackets

(a) a (4y – 3x) (b) (2x-1)(x+4) Q2. Calculate

Q3. Write down all the number that divide into 12 without leaving a remainder.

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Learning IntentionLearning Intention Success CriteriaSuccess Criteria

1.1. To identify the HCF for To identify the HCF for given terms.given terms.

1. We are learning 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.

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FactorisingFactorisingUsing FactorsNat 5

FactorisingFactorisingw

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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

FactorisingFactorisingw

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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

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FactorisingFactorising

Factorise the following :

(a) 3x + 6

(b) 4xy – 2x

(c) 6a + 7a2

(d) y2 - y

3(x + 2)

2x(2y – 1)

a(6 + 7a)

y(y – 1)

Be careful !

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Now try

N5 TJ Ex 7.1Ch7 (page 65)

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FactorsFactorsNat 5

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.com 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 operationassociated with the word ‘difference’.

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

1. We are learning 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.

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Difference of Difference of Two SquaresTwo SquaresNat 5

21 Apr 202321 Apr 2023 Created by Mr. Lafferty@www.mathsrevision.comCreated by Mr. Lafferty@www.mathsrevision.comww

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.com 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

Secondsquare term

Difference

21 Apr 202321 Apr 2023 Created by Mr. Lafferty@www.mathsrevision.comCreated by Mr. Lafferty@www.mathsrevision.comww

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.com 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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.com 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 )

Now try N5 TJ Ex 7.2 Q1

Ch7 (page 66)

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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

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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 )

Ch7 (page 66)

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.com Q1. Multiple out the bracket and simplify.

(a) y ( y + 6 ) -7y

Q2. Factorise 49 – 4x2

Q3. Write in scientific notation 0.0341

1.1. Understand the steps of Understand the steps of the St. Andrew’s Cross the St. Andrew’s Cross method.method.

2.2. Be able to factorise Be able to factorise quadratics using SAC quadratics using SAC method.method.

1. We are learning how to factorise trinomials ( quadratics) using

St. Andrew's Cross method.

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FactorisingFactorisingUsing St. Andrew’s Cross methodUsing St. Andrew’s Cross method

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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.

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(x + 1)(x + 2)

x2 + 2x

A LITTLE REVISIONMultiply out the brackets and Simplify

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

2. Tidy up !x2 + 3x + 2

RemovingRemovingDouble BracketsDouble Brackets

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(x + 1)(x + 2) x2 + 3x

We use SAC method to go the opposite way

+ 2FOIL

(x + 1)(x + 2) x2 + 3x + 2SAC

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+ 2+ 2

x2 + 3x + 2

Strategy for factorising quadratics

Find two numbers that multiply to give last number (+2)

and Diagonals sum to give middle value +3x.

( ) ( )

(+2) x( +1) = +2

(+2x) +( +1x) = +3x

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x2 + 6x + 5

( ) ( )

and Diagonals sum to give middle value +6x.

(+5) x( +1) = +5

(+5x) +( +1x) = +6x

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+ 4+ 4

x2 + x - 12

( ) ( )

One number must be + and one -

Find two numbers that multiply to give last number (-12)

and Diagonals sum to give middle value +x.

(+4) x( -3) = -12

(+4x) +( -3x) = +x

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- 2- 2

x2 - 4x + 4

( ) ( )

Both numbers must be -

and Diagonals sum to give middle value -4x.

(-2) x( -2) = -4

(-2x) +( -2x) = -4x

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- 3- 3

x2 - 2x - 3

( ) ( )

and Diagonals sum to give middle value -2x

(-3) x( +1) = -3

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

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Factorise using SAC method

(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 )

Now try N5 TJ Ex 7.3 Q1 & 2Ch7 (page 67)

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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

Q2. Factorise 2 – 3x – x2

1.1. Be able to factorise Be able to factorise trinomials / quadratics trinomials / quadratics using SAC.using SAC.

1. To show how to factorise trinomials ( quadratics) of the form ax2 + bx +c using SAC.

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- 43x - 4

3x2 - x - 4

( ) ( )

and Diagonals sum to give middle value -x

(-4) x( +1) = -4

(3x) +( -4x) = -x

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2x2 - x - 3

( ) ( )

and Diagonals sum to give middle value -x

(-3) x( +1) = -3

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

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4x2 - 4x - 3Two numbers that multiply to give

last number (-3)and

Diagonals sum to give middle value (-4x)

( ) ( )

Keeping the LHS fixed

Can we do it !

one number is + and

one number is -

Factors1 and -3-1 and 3

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4x2 - 4x - 3

( ) ( )

Find another set of factors for LHS

Factors1 and -3-1 and 3

Repeat the factors for RHS to see if it factorises now

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8x2+22x+15Find two numbers that

multiply to give last number (+15)and

Diagonals sum to give middle value (+22x)8x

( ) ( )

Keeping the LHS fixed

Can we do it !

Both numbers

must be +

Find all the factors of (+15) then try and

factorise

Factors1 and 153 and 5

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8x2+22x+15

( ) ( )

+ 3+ 3

Find another set of factors for LHS

Factors3 and 5

1 and 15

Repeat the factors for RHS to see if it factorises now

Now try N5 TJ Ex 7.3 Q1 & Q3Ch7 (page 68)

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.com Q1. Multiple out the brackets and simplify.

(a) ( 2x – 5 )( x + 5 )

Q3. Factorise 3ab – b2

Q2. After a 20% discount a watch is on sale for £240. What was the original price of the watch.

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.

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Summary ofSummary ofFactorisingFactorising

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Summary ofSummary ofFactorisingFactorisingNat 5

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.

Only TWO terms

THREE terms

Ch7 (page 68)

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If you can successfully complete this exercise

then you have the necessary skills to pass the algebraic part of the

course.

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