Www.mathsrevision.com S3 Credit Algebraic Operations 30-May-14Created by Mr....

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S3 Credit

Algebraic OperationsAlgebraic Operations

Apr 10, 2023Apr 10, 2023 Created by Mr. Created by Mr. [email protected]@mathsrevision.com

Difference of Squares

Factors / HCF

Common Factors

Factorising Trinomials (Quadratics)Factor Priority

Starter QuestionsStarter Questions

Apr 10, 2023Apr 10, 2023 Created by Mr. Created by Mr. [email protected]@mathsrevision.comw

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(a) a (4y – 3x) (b) (2x-1)(x+4) Q2. True or false

S3 Credit

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

5 1 4 11

6 3 3 3

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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.

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FactorsFactorsUsing FactorsS3

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FactorsFactorsw

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Factors

Example : Find the factors of 56.

F56 = 1 and 56

2 and 28

S3 Credit

Numbers that divide into 56 without leaving a remainder

4 and 14

7 and 8

FactorsFactors

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Highest Common Factor

We need to write out all factor pairs in order to find

the Highest Common Factor.

HighestCommonFactor

LargestSameNumber

S3 Credit

F8 = 1 and 8

2 and 4

FactorsFactors


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HCF = 4

F12 = 1 and 12

2 and 6

3 and 4

Highest Common FactorS3 Credit

2 and 5x

F4x =1, and 4x

FactorsFactors


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HCF = x

Fx2 = 1 and x2


F5 = 1 and 5

Example : Find the HCF of 5 and 10x.

HCF = 5

F10x = 1, and 10x

S3 Credit

5 and 2x

10 and x

2 and 2x

4 and x

x and x

F ab = 1 and ab

a and b

FactorsFactors


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HCF = b

F2b = 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

S3 Credit


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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

S3 Credit

10 Apr 202310 Apr 2023 Created by Mr. LaffertyCreated by Mr. Lafferty

Now try Ex 2.1 & 3.1

First Column in each QuestionCh5 (page 86)

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FactorsFactorsS3

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(a) a (4y – 3x) -2ay (b) (x + 5)(x - 5) Q2. Write out in full

S3 Credit

Q3. True or False all the factors of 5x2 are1, x, 5

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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.

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FactorisingFactorisingUsing FactorsS3

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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

S3 Credit


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

S3 Credit


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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 !

S3 Credit



First 2 Columns only

Ch5 (page 88)

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FactorisingFactorisingS3

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The sale price is £32. Show that the original price was £40

Q2. Factorise 3x2 – 6x

S3 Credit

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. 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.

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Difference of Difference of Two SquaresTwo SquaresS3

Credit


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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


Credit


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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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10 Apr 202310 Apr 2023 Created by Mr. LaffertyCreated by Mr. Laffertyww

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Format a2 – b2

Always the difference sign -

( a + b )( a – b )


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Factorise using the difference of two squares

(a) x2 – 72

(b) w2 – 1

(c) 9a2 – b2

(d) 16y2 – 100k2

(x + 7 )( x – 7 )

( w + 1 )( w – 1 )

( 3a + b )( 3a – b )

( 4y + 10k )( 4y – 10k )


Credit


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Trickier type of questions to factorise.Sometimes we need to take out a common factor

and then use the difference of two squares.

ExampleFactorise 2a2 - 18

2( a + 3 )( a – 3 )


Credit

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 )


Credit



First 2 Columns only

Ch5 (page 90)

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y ( y + 6 ) -7y = y2 -7y + 6

Q2. Fill in the ? 49 – 4x2 = ( ? + ?x)(? – 2?)

S3 Credit

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. To show how to factorise trinomials ( quadratics) using

St. Andrew's Cross method.

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S3 Credit

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


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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.


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S3 Credit

(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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S3 Credit

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

We use the SAC method to go the opposite way


+ 2FOIL

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


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S3 Credit

+ 1

+ 2+ 2


x2 + 3x + 2

Strategy for factorising quadratics


x

x + 1

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

and Diagonals sum to give middle value +3x.

( ) ( )

x

x

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

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

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S3 Credit

+ 5

+ 1

+ 5


x2 + 6x + 5



x

x + 1

( ) ( )

x

x


and Diagonals sum to give middle value +6x.

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

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

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S3 Credit

- 3

+ 4+ 4

- 3


x2 + x - 12



x

x

( ) ( )

x

x

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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S3 Credit

- 2

- 2- 2

- 2


x2 - 4x + 4



x

x

( ) ( )

x

x

Both numbers must be -


and Diagonals sum to give middle value -4x.

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

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

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S3 Credit

- 3- 3

+ 1


x2 - 2x - 3



x

x + 1

( ) ( )

x

x



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 + 6y + 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 )

S3 Credit



Now try Ex6.1

Ch5 (page 93)

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S3 Credit




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HP terms are 10% deposit the 6 months equal payments of £120. Show that you pay £90 using HP terms.

Q2. Factorise 2 + x – x2

S3 Credit



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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S3 Credit


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S3 Credit

- 43x - 4

+ 1


3x2 - x - 4



3x

x + 1

( ) ( )

x



and Diagonals sum to give middle value -x

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

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

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S3 Credit

- 32x

- 32x

+ 1


2x2 - x - 3



x + 1

( ) ( )

x



and Diagonals sum to give middle value -x

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

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

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S3 Credit

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

last number (-3)and

Diagonals sum to give middle value (-4x)

4x



x

( ) ( )

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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S3 Credit

- 3

+ 1

2x

- 3

4x2 - 4x - 3

2x



2x

( ) ( )

Find another set of factors for LHS

Factors1 and -3-1 and 3

Repeat the factors for RHS to see if it factorises now

+ 12x

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S3 Credit

8x2+22x+15Find two numbers that

multiply to give last number (+15)and

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



x

( ) ( )

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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S3 Credit

+ 5

8x2+22x+15

4x



2x

( ) ( )

+ 5

+ 3+ 3

4x2x

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 Ex 7.1First 2 columns

onlyCh5 (page 95)

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S3 Credit




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S3 Credit

Q3. True or false 3a2 b – ab2 =a2b2(3b – a)

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

Use a multiplication table to expand out (2x – 5)(x + 5)



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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S3 Credit

Summary ofSummary ofFactorisingFactorising


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Summary ofSummary ofFactorisingFactorisingS3

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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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Summary ofSummary ofFactorisingFactorisingS3

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Take Out Common Factor

2 squares

St. Andrew’s Cross methodSt. Andrew’s Cross method

Difference


Now try Ex 8.1

Ch5 (page 97)

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S3 Credit

Summary ofSummary ofFactorisingFactorising

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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