Created by Mr. Lafferty@mathsrevision.com Algebraic Operations Factors / HCF Common Factors Difference of Squares Factorising Trinomials (Quadratics) Factor Priority See Quadratic Theory for Exam Questions 3-Dec-17 Created by Mr. Lafferty@mathsrevision.com

Created by Mr. Lafferty@mathsrevision.com Starter Questions Nat 5 Q1. Remove the brackets (a) a (4y – 3x) (b) (2x-1)(x+4) Q2. Calculate www.mathsrevision.com Q3. Write down all the number that divide into 12 without leaving a remainder. 3-Dec-17 Created by Mr. Lafferty@mathsrevision.com

Created by Mr. Lafferty@www.mathsrevision.com Factorising Nat 5 Using Factors Learning Intention Success Criteria We are learning 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 3-Dec-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 Nat 5 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 ) 3-Dec-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 Nat 5 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 ) 3-Dec-17 Created by Mr. Lafferty@www.mathsrevision.com

Created by Mr. Lafferty@mathsrevision.com Factorising Nat 5 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) 3-Dec-17 Created by Mr. Lafferty@mathsrevision.com

Factors Now try N5 TJ Ex 7.1 Ch7 (page 65) www.mathsrevision.com Nat 5 3-Dec-17 Created by Mr. Lafferty

Created by Mr. Lafferty@mathsrevision.com Starter Questions Nat 5 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 www.mathsrevision.com Q3. Write down the arithmetic operation associated with the word ‘difference’. 3-Dec-17 Created by Mr. Lafferty@mathsrevision.com

Created by Mr. Lafferty@www.mathsrevision.com Difference of Two Squares Nat 5 Learning Intention Success Criteria We are learning 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. 3-Dec-17 Created by Mr. Lafferty@www.mathsrevision.com

Difference of Two Squares Nat 5 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 3-Dec-17 Created by Mr. Lafferty@www.mathsrevision.com

Difference of Two Squares Nat 5 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 - 3-Dec-17 Created by Mr. Lafferty@www.mathsrevision.com

Always the difference sign - Difference of Two Squares Nat 5 Keypoints Format a2 – b2 www.mathsrevision.com Always the difference sign - ( a + b )( a – b ) 3-Dec-17 Created by Mr. Lafferty

Difference of Two Squares Nat 5 Factorise using the difference of two squares (x + y )( x – y ) (a) x2 – y2 w2 – z2 9a2 – b2 (d) 16y2 – 100k2 ( w + z )( w – z ) www.mathsrevision.com ( 3a + b )( 3a – b ) ( 4y + 10k )( 4y – 10k ) 3-Dec-17 Created by Mr. Lafferty

Created by Mr. Lafferty@www.mathsrevision.com Difference of Two Squares Nat 5 Now try N5 TJ Ex 7.2 Q1 Ch7 (page 66) www.mathsrevision.com 3-Dec-17 Created by Mr. Lafferty@www.mathsrevision.com

Difference of Two Squares Nat 5 Trickier type of questions to factorise. Sometimes we need to take out a common And the 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 ) 3-Dec-17 Created by Mr. Lafferty

Difference of Two Squares Nat 5 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 ) 3-Dec-17 Created by Mr. Lafferty

Created by Mr. Lafferty@www.mathsrevision.com Difference of Two Squares Nat 5 Now try N5 TJ Ex 7.2 Q2 Ch7 (page 66) www.mathsrevision.com 3-Dec-17 Created by Mr. Lafferty@www.mathsrevision.com

Starter Questions Q1. Multiple out the bracket and simplify. Nat 5 Q1. Multiple out the bracket and simplify. (a) y ( y + 6 ) -7y Q2. Factorise 49 – 4x2 www.mathsrevision.com Q3. Write in scientific notation 0.0341 3-Dec-17 Created by Mr. Lafferty@mathsrevision.com

Using St. Andrew’s Cross method Factorising Using St. Andrew’s Cross method Nat 5 Learning Intention Success Criteria We are learning 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 3-Dec-17 Created by Mr. Lafferty@www.mathsrevision.com

Using St. Andrew’s Cross method Factorising Using St. Andrew’s Cross method Nat 5 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. www.mathsrevision.com 3-Dec-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 ! 3-Dec-17 Created by Mr. Lafferty@mathsrevision.com

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

Using St. Andrew’s Cross method Factorising Using St. Andrew’s Cross method Nat 5 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 ) 3-Dec-17 Created by Mr. Lafferty

Created by Mr. Lafferty@www.mathsrevision.com Difference of Two Squares Nat 5 Now try N5 TJ Ex 7.3 Q1 & 2 Ch7 (page 67) www.mathsrevision.com 3-Dec-17 Created by Mr. Lafferty@www.mathsrevision.com

Created by Mr. Lafferty@mathsrevision.com Starter Questions Nat 5 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 HP. www.mathsrevision.com Q2. Factorise 2 – 3x – x2 3-Dec-17 Created by Mr. Lafferty@mathsrevision.com

Using St. Andrew’s Cross method Factorising Using St. Andrew’s Cross method Nat 5 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 3-Dec-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 ( ) ( ) 3-Dec-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 ( ) ( ) 3-Dec-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 ! ( ) ( ) 3-Dec-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 ( ) ( ) 3-Dec-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 ! ( ) ( ) 3-Dec-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 ( ) ( ) 3-Dec-17 Created by Mr. Lafferty@mathsrevision.com

Created by Mr. Lafferty@www.mathsrevision.com Difference of Two Squares Nat 5 Now try N5 TJ Ex 7.3 Q1 & Q3 Ch7 (page 68) www.mathsrevision.com 3-Dec-17 Created by Mr. Lafferty@www.mathsrevision.com

Created by Mr. Lafferty@mathsrevision.com Starter Questions Nat 5 Q1. Multiple out the brackets and simplify. (a) ( 2x – 5 )( x + 5 ) Q2. After a 20% discount a watch is on sale for £240. What was the original price of the watch. www.mathsrevision.com Q3. Factorise 3ab – b2 3-Dec-17 Created by Mr. Lafferty@mathsrevision.com

Created by Mr. Lafferty@www.mathsrevision.com Summary of Factorising Nat 5 Learning Intention Success Criteria To explain the factorising priorities. Be able use the factorise priorities to factorise various expressions. www.mathsrevision.com 3-Dec-17 Created by Mr. Lafferty@www.mathsrevision.com

Created by Mr. Lafferty@www.mathsrevision.com Summary of Factorising Nat 5 When we are asked to factorise there is priority we must do it in. Only TWO terms 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. THREE terms 3-Dec-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. Difference of Two Squares Nat 5 Now try N5 TJ Ex 7.3 Q4 Ch7 (page 68) www.mathsrevision.com 3-Dec-17 Created by Mr. Lafferty@www.mathsrevision.com