Algebraic Operations www mathsrevision com Nat 5 Factors

Algebraic Operations www. mathsrevision. com Nat 5 Factors / HCF Common Factors Difference of Squares Factorising Trinomials (Quadratics) Factor Priority See Quadratic Theory for Exam Questions 27 -Nov-20 Created by Mr. Lafferty@mathsrevision. com

Starter Questions www. mathsrevision. com Nat 5 Q 1. Remove the brackets (a) a (4 y – 3 x) (b) (2 x-1)(x+4) Q 2. Calculate Q 3. Write down all the number that divide into 12 without leaving a remainder. 27 -Nov-20 Created by Mr. Lafferty@mathsrevision. com

Factorising www. mathsrevision. com Nat 5 Using Factors Learning Intention 1. We are learning how to factorise terms using the Highest Common Factor and one bracket term. 27 -Nov-20 Success Criteria 1. To identify the HCF for given terms. 2. Factorise terms using the HCF and one bracket term. Created by Mr. Lafferty@www. mathsrevision. com

Check by multiplying Factorising Nat 5 www. mathsrevision. com Example out the bracket to get back to where you Factorise 3 x +started 15 1. Find the HCF for 3 x and 15 2. HCF goes outside the bracket 3. To see what goes inside the bracket divide each term by HCF 3 x ÷ 3 = x 27 -Nov-20 15 ÷ 3 = 5 Created by Mr. Lafferty@www. mathsrevision. com 3 3( ) 3( x + 5 )

Check by multiplying Factorising Nat 5 www. mathsrevision. com Example out the bracket to get back to where you Factorise 4 x 2 –started 6 xy 1. Find the HCF for 4 x 2 and 6 xy 2. HCF goes outside the bracket 3. To see what goes inside the bracket divide each term by HCF 4 x 2 ÷ 2 x =2 x 27 -Nov-20 6 xy ÷ 2 x = 3 y Created by Mr. Lafferty@www. mathsrevision. com 2 x 2 x( ) 2 x( 2 x- 3 y )

Factorising Nat 5 www. mathsrevision. com Factorise the following : (a) 3(x + 2) 3 x + 6 Be careful ! 2 x(2 y – 1) (b) 4 xy – 2 x (c) 6 a + 7 a 2 a(6 + 7 a) (d) y 2 - y y(y – 1) 27 -Nov-20 Created by Mr. Lafferty@mathsrevision. com

Factors www. mathsrevision. com Nat 5 Now try N 5 TJ Ex 7. 1 Ch 7 (page 65) 27 -Nov-20 Created by Mr. Lafferty

Starter Questions www. mathsrevision. com Nat 5 Q 1. In a sale a jumper is reduced by 20%. The sale price is £ 32. What is the original price of the jumper. Q 2. Factorise 3 x 2 – 6 x Q 3. Write down the arithmetic operation associated with the word ‘difference’. 27 -Nov-20 Created by Mr. Lafferty@mathsrevision. com

Difference of Two Squares www. mathsrevision. com Nat 5 Learning Intention 1. We are learning how to factorise the special case of the difference of two squares. Success Criteria 1. Recognise when we have a difference of two squares. 2. Factorise the difference of two squares. 27 -Nov-20 Created by Mr. Lafferty@www. mathsrevision. com

Difference of Two Squares www. mathsrevision. com 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 a 2 – b 2 First square term 27 -Nov-20 Difference Second square term Created by Mr. Lafferty@www. mathsrevision. com

Difference of Two Squares www. mathsrevision. com Nat 5 2 by multiplying a 2 – out b. Check the bracket to get First square term back to where you Second Difference started square term This factorises to ( a + b )( a – b ) Two brackets the same except for + and a 27 -Nov-20 Created by Mr. Lafferty@www. mathsrevision. com

Difference of Two Squares www. mathsrevision. com Nat 5 Keypoints Format a 2 – b 2 Always the difference sign ( a + b )( a – b ) 27 -Nov-20 Created by Mr. Lafferty

Difference of Two Squares Nat 5 www. mathsrevision. com Factorise using the difference of two squares (a) x 2 – y 2 (x + y )( x – y ) (b) w 2 – z 2 ( w + z )( w – z ) (c) 9 a 2 – b 2 (d) 16 y 2 – 100 k 2 27 -Nov-20 ( 3 a + b )( 3 a – b ) ( 4 y + 10 k )( 4 y – 10 k ) Created by Mr. Lafferty

Difference of Two Squares www. mathsrevision. com Nat 5 Now try N 5 TJ Ex 7. 2 Q 1 Ch 7 (page 66) 27 -Nov-20 Created by Mr. Lafferty@www. mathsrevision. com

Difference of Two Squares www. mathsrevision. com 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 2 a 2 - 18 First take out common factor 2(a 2 - 9) Now apply the difference of two squares 2( a + 3 )( a – 3 ) 27 -Nov-20 Created by Mr. Lafferty

Difference of Two Squares Nat 5 www. mathsrevision. com Factorise these trickier expressions. (a) 6 x 2 – 24 6(x + 2 )( x – 2 ) (b) 3 w 2 – 3 3( w + 1 )( w – 1 ) (c) 8 – 2 b 2 (d) 27 w 2 – 12 27 -Nov-20 2( 2 + b )( 2 – b ) 3(3 w + 2 )( 3 w – 2 ) Created by Mr. Lafferty

Difference of Two Squares www. mathsrevision. com Nat 5 Now try N 5 TJ Ex 7. 2 Q 2 Ch 7 (page 66) 27 -Nov-20 Created by Mr. Lafferty@www. mathsrevision. com

Starter Questions Nat 5 www. mathsrevision. com Q 1. Multiple out the bracket and simplify. (a) y ( y + 6 ) -7 y Q 2. Factorise 49 – 4 x 2 Q 3. Write in scientific notation 0. 0341 27 -Nov-20 Created by Mr. Lafferty@mathsrevision. com

Factorising Using St. Andrew’s Cross method www. mathsrevision. com Nat 5 Learning Intention 1. We are learning how to factorise trinomials ( quadratics) using St. Andrew's Cross method. 27 -Nov-20 Success Criteria 1. Understand the steps of the St. Andrew’s Cross method. 2. Be able to factorise quadratics using SAC method. Created by Mr. Lafferty@www. mathsrevision. com

Factorising Using St. Andrew’s Cross method www. mathsrevision. com 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. 27 -Nov-20 Created by Mr. Lafferty@www. mathsrevision. com

Removing Double Brackets www. mathsrevision. com Nat 5 A LITTLE REVISION Multiply out the brackets and Simplify (x + 1)(x + 2) 1. Write down 2. Tidy up ! 27 -Nov-20 F O I L x 2 + 2 x + 2 x 2 + 3 x + 2 Created by Mr. Lafferty@mathsrevision. com

Factorising Using St. Andrew’s Cross method Nat 5 www. mathsrevision. com We use SAC method to go the opposite way (x + 1)(x + 2) 27 -Nov-20 FOIL SAC Created by Mr. Lafferty@mathsrevision. com x 2 + 3 x+ 2

Factorising Using St. Andrew’s Cross method Nat 5 www. mathsrevision. com Strategy for factorising quadratics Find two numbers that multiply to give last number (+2) and Diagonals sum to give middle value +3 x. x 2 + 3 x + 2 x +1 ( )( 27 -Nov-20 ) Created by Mr. Lafferty@mathsrevision. com (+2) x( +1) = +2 (+2 x) +( +1 x) = +3 x

Factorising Using St. Andrew’s Cross method Nat 5 www. mathsrevision. com Strategy for factorising quadratics x 2 + 6 x + 5 Find two numbers that multiply to give last number (+5) and Diagonals sum to give middle value +6 x x +5 x +1 ( 27 -Nov-20 )( ) Created by Mr. Lafferty@mathsrevision. com (+5) x( +1) = +5 (+5 x) +( +1 x) = +6 x

Factorising One number Using St. Andrew’s Cross method must be + and one - Nat 5 www. mathsrevision. com Strategy for factorising quadratics x 2 + x - 12 Find two numbers that multiply to give last number (-12) and Diagonals sum to give middle value +x. x +4 x -3 ( 27 -Nov-20 )( ) Created by Mr. Lafferty@mathsrevision. com (+4) x( -3) = -12 (+4 x) +( -3 x) = +x

Factorising Both numbers Using St. Andrew’s Cross method must be - Nat 5 www. mathsrevision. com Strategy for factorising quadratics x 2 - 4 x + 4 Find two numbers that multiply to give last number (+4) and Diagonals sum to give middle value -4 x. x -2 ( 27 -Nov-20 )( ) Created by Mr. Lafferty@mathsrevision. com (-2) x( -2) = -4 (-2 x) +( -2 x) = -4 x

Factorising One number Using St. Andrew’s Cross method must be + and one - Nat 5 www. mathsrevision. com Strategy for factorising quadratics x 2 - 2 x - 3 Find two numbers that multiply to give last number (-3) and Diagonals sum to give middle value -2 x x -3 x +1 ( 27 -Nov-20 )( ) Created by Mr. Lafferty@mathsrevision. com (-3) x( +1) = -3 (-3 x) +( x) = -2 x

Factorising Using St. Andrew’s Cross method Nat 5 www. mathsrevision. com Factorise using SAC method (a) m 2 + 2 m +1 (m + 1 )( m + 1 ) (b) y 2 + 6 y + 5 ( y + 5 )( y + 1 ) (c) b 2 – b - 2 ( b - 2 )( b + 1 ) (d) a 2 – 5 a + 6 ( a - 3 )( a – 2 ) 27 -Nov-20 Created by Mr. Lafferty

Difference of Two Squares www. mathsrevision. com Nat 5 Now try N 5 TJ Ex 7. 3 Q 1 & 2 Ch 7 (page 67) 27 -Nov-20 Created by Mr. Lafferty@www. mathsrevision. com

Starter Questions www. mathsrevision. com Nat 5 Q 1. 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. Q 2. Factorise 27 -Nov-20 2 – 3 x – x 2 Created by Mr. Lafferty@mathsrevision. com

Factorising Using St. Andrew’s Cross method www. mathsrevision. com Nat 5 Learning Intention 1. To show to factorise trinomials ( quadratics) of the form ax 2 + bx +c using SAC. 27 -Nov-20 Success Criteria 1. Be able to factorise trinomials / quadratics using SAC. Created by Mr. Lafferty@www. mathsrevision. com

Factorising One number Using St. Andrew’s Cross method must be + and one - Nat 5 www. mathsrevision. com Strategy for factorising quadratics 3 x 2 - x - 4 Find two numbers that multiply to give last number (-4) and Diagonals sum to give middle value -x 3 x -4 x +1 ( 27 -Nov-20 )( ) Created by Mr. Lafferty@mathsrevision. com (-4) x( +1) = -4 (3 x) +( -4 x) = -x

Factorising One number Using St. Andrew’s Cross method must be + and one - Nat 5 www. mathsrevision. com Strategy for factorising quadratics 2 x 2 - x - 3 Find two numbers that multiply to give last number (-3) and Diagonals sum to give middle value -x 2 x -3 x +1 ( 27 -Nov-20 )( ) Created by Mr. Lafferty@mathsrevision. com (-3) x( +1) = -3 (-3 x) +( +2 x) = -x

Factorisingone number is + Using St. Andrew’s Cross method and one number is - www. mathsrevision. com Nat 5 Two numbers that multiply to give last number (-3) and Diagonals sum to give middle value (-4 x) 4 x 2 - 4 x - 3 4 x Keeping the LHS fixed Factors 1 and -3 -1 and 3 x ( 27 -Nov-20 )( ) Can we do it ! Created by Mr. Lafferty@mathsrevision. com

Factorising Using St. Andrew’s Cross method www. mathsrevision. com Nat 5 Find another set of factors for LHS 4 x 2 - 4 x - 3 2 x -3 2 x +1 ( 27 -Nov-20 )( Repeat the factors for RHS to see if it factorises now ) Created by Mr. Lafferty@mathsrevision. com Factors 1 and -3 -1 and 3

Factorising Both numbers Using St. Andrew’s Cross method must be + www. mathsrevision. com Nat 5 Find two numbers that multiply to give last number (+15) and Diagonals sum to give middle value (+22 x) 8 x 2+22 x+15 8 x Keeping the LHS fixed Factors 1 and 15 factors Find all the 3 andtry 5 and factorise of (+15) then x ( 27 -Nov-20 )( ) Can we do it ! Created by Mr. Lafferty@mathsrevision. com

Factorising Using St. Andrew’s Cross method www. mathsrevision. com Nat 5 Find another set of factors for LHS 8 x 2+22 x+15 4 x +5 2 x +3 ( 27 -Nov-20 )( Repeat the factors for RHS to see if it factorises now ) Created by Mr. Lafferty@mathsrevision. com Factors 3 and 5 1 and 15

Difference of Two Squares www. mathsrevision. com Nat 5 Now try N 5 TJ Ex 7. 3 Q 1 & Q 3 Ch 7 (page 68) 27 -Nov-20 Created by Mr. Lafferty@www. mathsrevision. com

Starter Questions Nat 5 www. mathsrevision. com Q 1. Multiple out the brackets and simplify. (a) ( 2 x – 5 )( x + 5 ) Q 2. After a 20% discount a watch is on sale for £ 240. What was the original price of the watch. Q 3. Factorise 27 -Nov-20 3 ab – b 2 Created by Mr. Lafferty@mathsrevision. com

Summary of Factorising www. mathsrevision. com Nat 5 Learning Intention 1. To explain the factorising priorities. 27 -Nov-20 Success Criteria 1. Be able use the factorise priorities to factorise various expressions. Created by Mr. Lafferty@www. mathsrevision. com

Summary of Factorising www. mathsrevision. com Nat 5 When we are asked to factorise there is priority we must do it in. 1. Take any common factors out and put Onlythem TWO outside the brackets. terms 2. Check for the difference of two squares. 3. Factorise any quadratic expression left. THREE terms 27 -Nov-20 Created by Mr. Lafferty@www. mathsrevision. com

If you can successfully Difference of complete this exercise then you have the Two Squares necessary skills to pass Nat 5 www. mathsrevision. com the algebraic part of the course. Now try N 5 TJ Ex 7. 3 Q 4 Ch 7 (page 68) 27 -Nov-20 Created by Mr. Lafferty@www. mathsrevision. com