Created by Mr. Lafferty@mathsrevision.com Algebraic Operations Factors / HCF Common Factors Difference of Squares Factorising Trinomials (Quadratics) Factor Priority 31-Mar-17 Created by Mr. Lafferty@mathsrevision.com

Created by Mr. Lafferty@mathsrevision.com Starter Questions S3 Credit Q1. Multiply out (a) a (4y – 3x) (b) (2x-1)(x+4) Q2. True or false www.mathsrevision.com Q3. Write down all the number that divide into 12 without leaving a remainder. 31-Mar-17 Created by Mr. Lafferty@mathsrevision.com

Created by Mr. Lafferty@mathsrevision.com Factors S3 Credit Using Factors Learning Intention Success Criteria To explain that a factor divides into a number without leaving a remainder To explain how to find Highest Common Factors To identify factors using factor pairs Find HCF for two numbers by comparing factors. www.mathsrevision.com 31-Mar-17 Created by Mr. Lafferty@mathsrevision.com

Created by Mr. Lafferty@mathsrevision.com Factors 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 www.mathsrevision.com 4 and 14 7 and 8 31-Mar-17 Created by Mr. Lafferty@mathsrevision.com

Factors Highest Common Factor Largest Same Number Credit Highest Common Factor Highest Common Factor Largest Same Number www.mathsrevision.com We need to write out all factor pairs in order to find the Highest Common Factor. 31-Mar-17 Created by Mr. Lafferty@mathsrevision.com

Created by Mr. Lafferty@mathsrevision.com Factors S3 Credit Highest Common Factor Example : Find the HCF of 8 and 12. F8 = 1 and 8 2 and 4 F12 = 1 and 12 2 and 6 3 and 4 www.mathsrevision.com HCF = 4 31-Mar-17 Created by Mr. Lafferty@mathsrevision.com

Created by Mr. Lafferty@mathsrevision.com Factors S3 Credit Highest Common Factor Example : Find the HCF of 4x and x2. F4x = 1, and 4x Fx2 = 1 and x2 2 and 2x x and x 4 and x HCF = x www.mathsrevision.com Example : Find the HCF of 5 and 10x. F5 = 1 and 5 F10x = 1, and 10x 2 and 5x HCF = 5 5 and 2x 31-Mar-17 Created by Mr. Lafferty@mathsrevision.com 10 and x

Created by Mr. Lafferty@mathsrevision.com Factors S3 Credit Highest Common Factor Example : Find the HCF of ab and 2b. F ab = 1 and ab a and b F2b = 1 and 2b 2 and b HCF = b www.mathsrevision.com Example : Find the HCF of 2h2 and 4h. F 2h2 = 1 and 2h2 2 and h2 , h and 2h F4h = 1 and 4h 2 and 2h 4 and h HCF = 2h 31-Mar-17 Created by Mr. Lafferty@mathsrevision.com

Created by Mr. Lafferty@mathsrevision.com Factors S3 Credit Find the HCF for these terms 8w (a) 16w and 24w 9y2 and 6y (c) 4h and 12h2 (d) ab2 and a2b 3y www.mathsrevision.com 4h ab 31-Mar-17 Created by Mr. Lafferty@mathsrevision.com

First Column in each Question Factors S3 Credit Now try Ex 2.1 & 3.1 First Column in each Question Ch5 (page 86) www.mathsrevision.com 31-Mar-17 Created by Mr. Lafferty

Created by Mr. Lafferty@mathsrevision.com Starter Questions S3 Credit Q1. Expand out (a) a (4y – 3x) -2ay (b) (x + 5)(x - 5) Q2. Write out in full www.mathsrevision.com Q3. True or False all the factors of 5x2 are 1, x, 5 31-Mar-17 Created by Mr. Lafferty@mathsrevision.com

Created by Mr. Lafferty@www.mathsrevision.com Factorising S3 Credit Using Factors Learning Intention Success Criteria To show how to factorise terms using the Highest Common Factor and one bracket term. To identify the HCF for given terms. Factorise terms using the HCF and one bracket term. www.mathsrevision.com 31-Mar-17 Created by Mr. Lafferty@www.mathsrevision.com

Factorising Example Factorise 3x + 15 1. Find the HCF for 3x and 15 3 Check by multiplying out the bracket to get back to where you started Factorising S3 Credit Example Factorise 3x + 15 1. Find the HCF for 3x and 15 3 2. HCF goes outside the bracket 3( ) www.mathsrevision.com To see what goes inside the bracket divide each term by HCF 3x ÷ 3 = x 15 ÷ 3 = 5 3( x + 5 ) 31-Mar-17 Created by Mr. Lafferty@www.mathsrevision.com

Factorising Example Factorise 4x2 – 6xy Check by multiplying out the bracket to get back to where you started Factorising S3 Credit Example Factorise 4x2 – 6xy 1. Find the HCF for 4x2 and 6xy 2x 2. HCF goes outside the bracket 2x( ) www.mathsrevision.com To see what goes inside the bracket divide each term by HCF 4x2 ÷ 2x =2x 6xy ÷ 2x = 3y 2x( 2x- 3y ) 31-Mar-17 Created by Mr. Lafferty@www.mathsrevision.com

Created by Mr. Lafferty@mathsrevision.com Factorising S3 Credit Factorise the following : 3(x + 2) (a) 3x + 6 4xy – 2x 6a + 7a2 (d) y2 - y Be careful ! 2x(2y – 1) www.mathsrevision.com a(6 + 7a) y(y – 1) 31-Mar-17 Created by Mr. Lafferty@mathsrevision.com

Created by Mr. Lafferty@www.mathsrevision.com Factorising S3 Credit Now try Ex 4.1 & 4.2 First 2 Columns only Ch5 (page 88) www.mathsrevision.com 31-Mar-17 Created by Mr. Lafferty@www.mathsrevision.com

Created by Mr. Lafferty@mathsrevision.com Starter Questions S3 Credit 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 www.mathsrevision.com Q3. Write down the arithmetic operation associated with the word ‘difference’. 31-Mar-17 Created by Mr. Lafferty@mathsrevision.com

Created by Mr. Lafferty@www.mathsrevision.com Difference of Two Squares S3 Credit Learning Intention Success Criteria To show how to factorise the special case of the difference of two squares. Recognise when we have a difference of two squares. www.mathsrevision.com Factorise the difference of two squares. 31-Mar-17 Created by Mr. Lafferty@www.mathsrevision.com

Difference of Two Squares 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 www.mathsrevision.com a2 – b2 First square term Second square term Difference 31-Mar-17 Created by Mr. Lafferty@www.mathsrevision.com

Difference of Two Squares Credit Check by multiplying out the bracket to get back to where you started a2 – b2 First square term Second square term Difference This factorises to www.mathsrevision.com ( a + b )( a – b ) Two brackets the same except for + and a - 31-Mar-17 Created by Mr. Lafferty@www.mathsrevision.com

Always the difference sign - Difference of Two Squares S3 Credit Keypoints Format a2 – b2 www.mathsrevision.com Always the difference sign - ( a + b )( a – b ) 31-Mar-17 Created by Mr. Lafferty

Difference of Two Squares Credit Factorise using the difference of two squares (x + 7 )( x – 7 ) (a) x2 – 72 w2 – 1 9a2 – b2 (d) 16y2 – 100k2 ( w + 1 )( w – 1 ) www.mathsrevision.com ( 3a + b )( 3a – b ) ( 4y + 10k )( 4y – 10k ) 31-Mar-17 Created by Mr. Lafferty

Difference of Two Squares 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) www.mathsrevision.com Now apply the difference of two squares 2( a + 3 )( a – 3 ) 31-Mar-17 Created by Mr. Lafferty

Difference of Two Squares Credit Factorise these trickier expressions. 6(x + 2 )( x – 2 ) (a) 6x2 – 24 3w2 – 3 8 – 2b2 (d) 27w2 – 12 3( w + 1 )( w – 1 ) www.mathsrevision.com 2( 2 + b )( 2 – b ) 3(3 w + 2 )( 3w – 2 ) 31-Mar-17 Created by Mr. Lafferty

Created by Mr. Lafferty@www.mathsrevision.com Difference of Two Squares S3 Credit Now try Ex 5.1 & 5.2 First 2 Columns only Ch5 (page 90) www.mathsrevision.com 31-Mar-17 Created by Mr. Lafferty@www.mathsrevision.com

Starter Questions Q1. True or false y ( y + 6 ) -7y = y2 -7y + 6 Credit Q1. True or false y ( y + 6 ) -7y = y2 -7y + 6 Q2. Fill in the ? 49 – 4x2 = ( ? + ?x)(? – 2?) www.mathsrevision.com Q3. Write in scientific notation 0.0341 31-Mar-17 Created by Mr. Lafferty@mathsrevision.com

Using St. Andrew’s Cross method Factorising Using St. Andrew’s Cross method S3 Credit Learning Intention Success Criteria To show how to factorise trinomials ( quadratics) using St. Andrew's Cross method. Understand the steps of the St. Andrew’s Cross method. 2. Be able to factorise quadratics using SAC method. www.mathsrevision.com 31-Mar-17 Created by Mr. Lafferty@www.mathsrevision.com

Using St. Andrew’s Cross method Factorising Using St. Andrew’s Cross method 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. www.mathsrevision.com 31-Mar-17 Created by Mr. Lafferty@www.mathsrevision.com

Created by Mr. Lafferty@mathsrevision.com Removing Double Brackets A LITTLE REVISION Multiply out the brackets and Simplify (x + 1)(x + 2) 1. Write down F O I L x2 + 2x + x + 2 x2 + 3x + 2 2. Tidy up ! 31-Mar-17 Created by Mr. Lafferty@mathsrevision.com

Using St. Andrew’s Cross method Factorising Using St. Andrew’s Cross method We use the SAC method to go the opposite way FOIL (x + 1)(x + 2) x2 + 3x + 2 SAC (x + 1)(x + 2) x2 + 3x + 2 31-Mar-17 Created by Mr. Lafferty@mathsrevision.com

Using St. Andrew’s Cross method Factorising Using St. Andrew’s Cross method 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 x + 2 + 2 (+2) x( +1) = +2 x x + 1 + 1 (+2x) +( +1x) = +3x ( ) ( ) 31-Mar-17 Created by Mr. Lafferty@mathsrevision.com

Using St. Andrew’s Cross method Factorising Using St. Andrew’s Cross method Strategy for factorising quadratics Find two numbers that multiply to give last number (+5) and Diagonals sum to give middle value +6x. x2 + 6x + 5 x x + 5 + 5 (+5) x( +1) = +5 x x + 1 + 1 (+5x) +( +1x) = +6x ( ) ( ) 31-Mar-17 Created by Mr. Lafferty@mathsrevision.com

Using St. Andrew’s Cross method One number must be + and one - Factorising Using St. Andrew’s Cross method Strategy for factorising quadratics Find two numbers that multiply to give last number (-12) and Diagonals sum to give middle value +x. x2 + x - 12 x x + 4 + 4 (+4) x( -3) = -12 x x - 3 - 3 (+4x) +( -3x) = +x ( ) ( ) 31-Mar-17 Created by Mr. Lafferty@mathsrevision.com

Using St. Andrew’s Cross method Both numbers must be - Factorising Using St. Andrew’s Cross method Strategy for factorising quadratics Find two numbers that multiply to give last number (+4) and Diagonals sum to give middle value -4x. x2 - 4x + 4 x x - 2 - 2 (-2) x( -2) = +4 x x - 2 - 2 (-2x) +( -2x) = -4x ( ) ( ) 31-Mar-17 Created by Mr. Lafferty@mathsrevision.com

Using St. Andrew’s Cross method One number must be + and one - Factorising Using St. Andrew’s Cross method Strategy for factorising quadratics Find two numbers that multiply to give last number (-3) and Diagonals sum to give middle value -2x x2 - 2x - 3 x x - 3 - 3 (-3) x( +1) = -3 x x + 1 + 1 (-3x) +( x) = -2x ( ) ( ) 31-Mar-17 Created by Mr. Lafferty@mathsrevision.com

Using St. Andrew’s Cross method Factorising Using St. Andrew’s Cross method S3 Credit Factorise using SAC method (m + 1 )( m + 1 ) (a) m2 + 2m + 1 y2 + 6y + 5 b2 – b - 2 (d) a2 – 5a + 6 ( y + 5 )( y + 1 ) www.mathsrevision.com ( b - 2 )( b + 1 ) ( a - 3 )( a – 2 ) 31-Mar-17 Created by Mr. Lafferty

Using St. Andrew’s Cross method Factorising Using St. Andrew’s Cross method S3 Credit Now try Ex6.1 Ch5 (page 93) www.mathsrevision.com 31-Mar-17 Created by Mr. Lafferty@www.mathsrevision.com

Created by Mr. Lafferty@mathsrevision.com Starter Questions 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. www.mathsrevision.com Q2. Factorise 2 + x – x2 31-Mar-17 Created by Mr. Lafferty@mathsrevision.com

Using St. Andrew’s Cross method Factorising Using St. Andrew’s Cross method S3 Credit Learning Intention Success Criteria To show how to factorise trinomials ( quadratics) of the form ax2 + bx +c using SAC. Be able to factorise trinomials / quadratics using SAC. www.mathsrevision.com 31-Mar-17 Created by Mr. Lafferty@www.mathsrevision.com

Using St. Andrew’s Cross method One number must be + and one - Factorising Using St. Andrew’s Cross method Strategy for factorising quadratics Find two numbers that multiply to give last number (-4) and Diagonals sum to give middle value -x 3x2 - x - 4 3x 3x - 4 - 4 (-4) x( +1) = -4 x x + 1 + 1 (3x) +( -4x) = -x ( ) ( ) 31-Mar-17 Created by Mr. Lafferty@mathsrevision.com

Using St. Andrew’s Cross method One number must be + and one - Factorising Using St. Andrew’s Cross method Strategy for factorising quadratics Find two numbers that multiply to give last number (-3) and Diagonals sum to give middle value -x 2x2 - x - 3 2x 2x - 3 - 3 (-3) x( +1) = -3 x x + 1 + 1 (-3x) +( +2x) = -x ( ) ( ) 31-Mar-17 Created by Mr. Lafferty@mathsrevision.com

Using St. Andrew’s Cross method one number is + and one number is - Factorising Using St. Andrew’s Cross method 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 Can we do it ! ( ) ( ) 31-Mar-17 Created by Mr. Lafferty@mathsrevision.com

Using St. Andrew’s Cross method Factorising Using St. Andrew’s Cross method Find another set of factors for LHS 4x2 - 4x - 3 Repeat the factors for RHS to see if it factorises now 2x 2x - 3 - 3 Factors 1 and -3 -1 and 3 2x 2x + 1 + 1 ( ) ( ) 31-Mar-17 Created by Mr. Lafferty@mathsrevision.com

Using St. Andrew’s Cross method Both numbers must be + Factorising Using St. Andrew’s Cross method 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 3 and 5 Find all the factors of (+15) then try and factorise x Can we do it ! ( ) ( ) 31-Mar-17 Created by Mr. Lafferty@mathsrevision.com

Using St. Andrew’s Cross method Factorising Using St. Andrew’s Cross method Find another set of factors for LHS 8x2+22x+15 Repeat the factors for RHS to see if it factorises now 4x 4x + 5 + 5 Factors 3 and 5 1 and 15 2x 2x + 3 + 3 ( ) ( ) 31-Mar-17 Created by Mr. Lafferty@mathsrevision.com

Using St. Andrew’s Cross method Factorising Using St. Andrew’s Cross method S3 Credit Now try Ex 7.1 First 2 columns only Ch5 (page 95) www.mathsrevision.com 31-Mar-17 Created by Mr. Lafferty@www.mathsrevision.com

Starter Questions S3 Credit Use a multiplication table to expand out (2x – 5)(x + 5) Q1. Q2. After a 20% discount a watch is on sale for £240. What was the original price of the watch. www.mathsrevision.com Q3. True or false 3a2 b – ab2 =a2b2(3b – a) 31-Mar-17 Created by Mr. Lafferty@mathsrevision.com

Created by Mr. Lafferty@www.mathsrevision.com Summary of Factorising S3 Credit Learning Intention Success Criteria To explain the factorising priorities. Be able use the factorise priorities to factorise various expressions. www.mathsrevision.com 31-Mar-17 Created by Mr. Lafferty@www.mathsrevision.com

Created by Mr. Lafferty@www.mathsrevision.com Summary of Factorising S3 Credit When we are asked to factorise there is priority we must do it in. Take any common factors out and put them outside the brackets. 2. Check for the difference of two squares. www.mathsrevision.com 3. Factorise any quadratic expression left. 31-Mar-17 Created by Mr. Lafferty@www.mathsrevision.com

2 Summary of Factorising www.mathsrevision.com squares Credit St. Andrew’s Cross method 2 squares Difference www.mathsrevision.com Take Out Common Factor 31-Mar-17 Created by Mr. Lafferty@www.mathsrevision.com

Created by Mr. Lafferty@www.mathsrevision.com If you can successfully complete this exercise then you have the necessary skills to pass the algebraic part of the course. Summary of Factorising S3 Credit Now try Ex 8.1 Ch5 (page 97) www.mathsrevision.com 31-Mar-17 Created by Mr. Lafferty@www.mathsrevision.com