Algebraic Operations S 3 Credit www mathsrevision com

Algebraic Operations S 3 Credit www. mathsrevision. com Factors / HCF Common Factors Difference of Squares Factorising Trinomials (Quadratics) Factor Priority 15 -Dec-21 Created by Mr. Lafferty@mathsrevision. com

Starter Questions www. mathsrevision. com S 3 Credit 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. 15 -Dec-21 Created by Mr. Lafferty@mathsrevision. com

Factors www. mathsrevision. com S 3 Credit Using Factors Learning Intention 1. To explain that a factor divides into a number without leaving a remainder 2. To explain how to find Highest Common Factors 15 -Dec-21 Success Criteria 1. To identify factors using factor pairs 2. Find HCF for two numbers by comparing factors. Created by Mr. Lafferty@mathsrevision. com

Factors www. mathsrevision. com S 3 Credit Factors Example : Find the factors of 56. Always divide by 1 and find its pair F 56 = 1 and 56 From 2 find other factors and their pairs 2 and 28 4 and 14 7 and 15 -Dec-21 8 Created by Mr. Lafferty@mathsrevision. com

Factors www. mathsrevision. com S 3 Credit Highest Common Factor Largest Same Number We need to write out all factor pairs in order to find the Highest Common Factor. 15 -Dec-21 Created by Mr. Lafferty@mathsrevision. com

Factors www. mathsrevision. com S 3 Credit Highest Common Factor Example : Find the HCF of 8 and 12. F 8 = 1 and 8 F 12 = 1 and 12 2 and 4 2 and 6 3 and 4 HCF = 4 15 -Dec-21 Created by Mr. Lafferty@mathsrevision. com

Factors www. mathsrevision. com S 3 Credit Highest Common Factor Example : Find the HCF of 4 x and x 2. F 4 x = 1, and 4 x , Fx 2 = 1 and x 2 2 and 2 x 4 and Example : x and x x HCF = x Find the HCF of 5 and 10 x. F 5 = 1 and 5 F 10 x = 1, and 10 x HCF = 5 15 -Dec-21 2 and 5 x , 5 and 2 x Created by Mr. Lafferty@mathsrevision. com 10 and x

Factors www. mathsrevision. com S 3 Credit Highest Common Factor Example : Find the HCF of ab and 2 b. F ab = 1 and ab F 2 b = 1 and 2 b a and b Example : Find the HCF of 2 h 2 and 4 h. F 2 h 2 = 1 and 2 h 2 2 and h 2 , h and 2 h 15 -Dec-21 2 and F 4 h = 1 and 4 h HCF = 2 h Created by Mr. Lafferty@mathsrevision. com 2 and 2 h 4 and h b HCF = b

Factors S 3 Credit www. mathsrevision. com Find the HCF for these terms (a) 16 w and 24 w 8 w (b) 9 y 2 and 6 y 3 y (c) 4 h and 12 h 2 4 h (d) ab 2 and a 2 b ab 15 -Dec-21 Created by Mr. Lafferty@mathsrevision. com

Factors www. mathsrevision. com S 3 Credit Now try Ex 2. 1 & 3. 1 First Column in each Question Ch 5 (page 86) 15 -Dec-21 Created by Mr. Lafferty

Starter Questions S 3 Credit www. mathsrevision. com Q 1. Remove the brackets (a) a (4 y – 3 x) -2 ay (b) Q 2. Write out in full Q 3. Find the factors of 5 x 2 15 -Dec-21 Created by Mr. Lafferty@mathsrevision. com (x + 5)(x - 5)

Factorising www. mathsrevision. com S 3 Credit Using Factors Learning Intention 1. To show to factorise terms using the Highest Common Factor and one bracket term. 15 -Dec-21 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 S 3 Credit 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 15 -Dec-21 15 ÷ 3 = 5 Created by Mr. Lafferty@www. mathsrevision. com 3 3( ) 3( x + 5 )

Check by multiplying Factorising S 3 Credit 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 15 -Dec-21 6 xy ÷ 2 x = 3 y Created by Mr. Lafferty@www. mathsrevision. com 2 x 2 x( ) 2 x( 2 x- 3 y )

Factorising S 3 Credit 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) 15 -Dec-21 Created by Mr. Lafferty@mathsrevision. com

Factorising www. mathsrevision. com S 3 Credit Now try Ex 4. 1 & 4. 2 First 2 Columns only Ch 5 (page 88) 15 -Dec-21 Created by Mr. Lafferty@www. mathsrevision. com

Starter Questions www. mathsrevision. com S 3 Credit 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’. 15 -Dec-21 Created by Mr. Lafferty@mathsrevision. com

Difference of Two Squares www. mathsrevision. com S 3 Credit Learning Intention 1. To show 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. 15 -Dec-21 Created by Mr. Lafferty@www. mathsrevision. com

Difference of Two Squares www. mathsrevision. com S 3 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 a 2 – b 2 First square term 15 -Dec-21 Difference Second square term Created by Mr. Lafferty@www. mathsrevision. com

Difference of Two Squares www. mathsrevision. com S 3 Credit 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 15 -Dec-21 Created by Mr. Lafferty@www. mathsrevision. com

Difference of Two Squares www. mathsrevision. com S 3 Credit Keypoints Format a 2 – b 2 Always the difference sign ( a + b )( a – b ) 15 -Dec-21 Created by Mr. Lafferty

Difference of Two Squares S 3 Credit 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 15 -Dec-21 ( 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 S 3 Credit 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 ) 15 -Dec-21 Created by Mr. Lafferty

Difference of Two Squares S 3 Credit 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 15 -Dec-21 2( 2 + b )( 2 – b ) 3(3 w + 2 )( 3 w – 2 ) Created by Mr. Lafferty

Difference of Two Squares www. mathsrevision. com S 3 Credit Now try Ex 5. 1 & 5. 2 First 2 Columns only Ch 5 (page 90) 15 -Dec-21 Created by Mr. Lafferty@www. mathsrevision. com

Starter Questions S 3 Credit 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 15 -Dec-21 Created by Mr. Lafferty@mathsrevision. com

Factorising Using St. Andrew’s Cross method www. mathsrevision. com S 3 Credit Learning Intention 1. To show to factorise trinomials ( quadratics) using St. Andrew's Cross method. 15 -Dec-21 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 S 3 Credit 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. 15 -Dec-21 Created by Mr. Lafferty@www. mathsrevision. com

Removing Double Brackets www. mathsrevision. com S 3 Credit A LITTLE REVISION Multiply out the brackets and Simplify (x + 1)(x + 2) 1. Write down 2. Tidy up ! 15 -Dec-21 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 S 3 Credit www. mathsrevision. com We use SAC method to go the opposite way (x + 1)(x + 2) 15 -Dec-21 FOIL SAC Created by Mr. Lafferty@mathsrevision. com x 2 + 3 x+ 2

Factorising Using St. Andrew’s Cross method S 3 Credit 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 ( )( 15 -Dec-21 ) Created by Mr. Lafferty@mathsrevision. com (+2) x( +1) = +2 (+2 x) +( +1 x) = +3 x

Factorising Using St. Andrew’s Cross method S 3 Credit 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 ( 15 -Dec-21 )( ) 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 - S 3 Credit 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 ( 15 -Dec-21 )( ) 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 - S 3 Credit 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 ( 15 -Dec-21 )( ) 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 - S 3 Credit 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 ( 15 -Dec-21 )( ) Created by Mr. Lafferty@mathsrevision. com (-3) x( +1) = -3 (-3 x) +( x) = -2 x

Factorising Using St. Andrew’s Cross method S 3 Credit www. mathsrevision. com Factorise using SAC method (a) m 2 + 2 m +1 (m + 1 )( m + 1 ) (b) y 2 + 6 m + 5 ( y + 5 )( y + 1 ) (c) b 2 – b -2 ( b - 2 )( b + 1 ) (d) a 2 – 5 a + 6 ( a - 3 )( a – 2 ) 15 -Dec-21 Created by Mr. Lafferty

Factorising Using St. Andrew’s Cross method www. mathsrevision. com S 3 Credit Now try Ex 6. 1 Ch 5 (page 93) 15 -Dec-21 Created by Mr. Lafferty@www. mathsrevision. com

Starter Questions www. mathsrevision. com S 3 Credit 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 15 -Dec-21 2 – 3 x – x 2 Created by Mr. Lafferty@mathsrevision. com

Factorising Using St. Andrew’s Cross method www. mathsrevision. com S 3 Credit Learning Intention 1. To show to factorise trinomials ( quadratics) of the form ax 2 + bx +c using SAC. 15 -Dec-21 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 - S 3 Credit 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 ( 15 -Dec-21 )( ) 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 - S 3 Credit 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 ( 15 -Dec-21 )( ) 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 S 3 Credit 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 ( 15 -Dec-21 )( ) Can we do it ! Created by Mr. Lafferty@mathsrevision. com

Factorising Using St. Andrew’s Cross method www. mathsrevision. com S 3 Credit Find another set of factors for LHS 4 x 2 - 4 x - 3 2 x -3 2 x +1 ( 15 -Dec-21 )( 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 S 3 Credit 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 ( 15 -Dec-21 )( ) Can we do it ! Created by Mr. Lafferty@mathsrevision. com

Factorising Using St. Andrew’s Cross method www. mathsrevision. com S 3 Credit Find another set of factors for LHS 8 x 2+22 x+15 4 x +5 2 x +3 ( 15 -Dec-21 )( Repeat the factors for RHS to see if it factorises now ) Created by Mr. Lafferty@mathsrevision. com Factors 3 and 5 1 and 15

Factorising Using St. Andrew’s Cross method www. mathsrevision. com S 3 Credit Now try Ex 7. 1 First 2 columns only Ch 5 (page 95) 15 -Dec-21 Created by Mr. Lafferty@www. mathsrevision. com

Starter Questions S 3 Credit 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 15 -Dec-21 3 ab – b 2 Created by Mr. Lafferty@mathsrevision. com

Summary of Factorising www. mathsrevision. com S 3 Credit Learning Intention 1. To explain the factorising priorities. 15 -Dec-21 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 S 3 Credit When we are asked to factorise there is priority we must do it in. 1. Take any common factors out and put them outside the brackets. 2. Check for the difference of two squares. 3. 15 -Dec-21 Factorise any quadratic expression left. Created by Mr. Lafferty@www. mathsrevision. com

Summary of Factorising www. mathsrevision. com S 3 Credit St. Andrew’s Cross method 2 Difference squares Take Out Common Factor 15 -Dec-21 Created by Mr. Lafferty@www. mathsrevision. com

If youof can successfully Summary complete this exercise then you have the Factorising necessary skills to pass www. mathsrevision. com S 3 Credit the algebraic part of the course. Now try Ex 8. 1 Ch 5 (page 97) 15 -Dec-21 Created by Mr. Lafferty@www. mathsrevision. com