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Algebraic Operations

Algebraic Operations. Factors / HCF. Common Factors. Difference of Squares. Factorising Trinomials (Quadratics). Factor Priority. S3 Credit. Q1. Multiply out (a) a (4y – 3x) (b) (2x-1)(x+4). Starter Questions. Q2. True or false. www.mathsrevision.com.

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Algebraic Operations

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

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

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

  4. 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 Created by Mr. Lafferty@mathsrevision.com

  5. Factors S3 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. Created by Mr. Lafferty@mathsrevision.com

  6. 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 Created by Mr. Lafferty@mathsrevision.com

  7. 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 Created by Mr. Lafferty@mathsrevision.com 10 and x

  8. 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 Created by Mr. Lafferty@mathsrevision.com

  9. 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 Created by Mr. Lafferty@mathsrevision.com

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

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

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

  13. Check by multiplying out the bracket to get back to where you started Factorising S3 Credit Factorise 3x + 15 Example 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 ) Created by Mr. Lafferty@www.mathsrevision.com

  14. Check by multiplying out the bracket to get back to where you started Factorising S3 Credit Factorise 4x2 – 6xy Example 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 ) Created by Mr. Lafferty@www.mathsrevision.com

  15. 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) Created by Mr. Lafferty@mathsrevision.com

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

  17. S3 Credit Q1. In a sale a jumper is reduced by 20%. The sale price is £32. Show that the original price was £40 Starter Questions Q2. Factorise 3x2 – 6x www.mathsrevision.com Q3. Write down the arithmetic operation associated with the word ‘difference’. Created by Mr. Lafferty@mathsrevision.com

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

  19. Difference of Two Squares S3 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 Created by Mr. Lafferty@www.mathsrevision.com

  20. Difference of Two Squares S3 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 - Created by Mr. Lafferty@www.mathsrevision.com

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

  22. Difference of Two Squares S3 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 ) Created by Mr. Lafferty

  23. Difference of Two Squares S3 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 2(a2 - 9) First take out common factor www.mathsrevision.com Now apply the difference of two squares 2( a + 3 )( a – 3 ) Created by Mr. Lafferty

  24. Difference of Two Squares S3 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 ) Created by Mr. Lafferty

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

  26. S3 Credit Q1. True or false y ( y + 6 ) -7y = y2 -7y + 6 Starter Questions Q2. Fill in the ? 49 – 4x2 = ( ? + ?x)(? – 2?) www.mathsrevision.com Q3. Write in scientific notation 0.0341 Created by Mr. Lafferty@mathsrevision.com

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

  28. 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 Created by Mr. Lafferty@www.mathsrevision.com

  29. 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 ! Created by Mr. Lafferty@mathsrevision.com

  30. 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 Created by Mr. Lafferty@mathsrevision.com

  31. 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 ( ) ( ) Created by Mr. Lafferty@mathsrevision.com

  32. 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 ( ) ( ) Created by Mr. Lafferty@mathsrevision.com

  33. 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 ( ) ( ) Created by Mr. Lafferty@mathsrevision.com

  34. 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 ( ) ( ) Created by Mr. Lafferty@mathsrevision.com

  35. 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 ( ) ( ) Created by Mr. Lafferty@mathsrevision.com

  36. 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 ) Created by Mr. Lafferty

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

  38. S3 Credit Starter Questions 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 Created by Mr. Lafferty@mathsrevision.com

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

  40. 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 ( ) ( ) Created by Mr. Lafferty@mathsrevision.com

  41. 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 ( ) ( ) Created by Mr. Lafferty@mathsrevision.com

  42. 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 Factors 1 and -3 -1 and 3 Keeping the LHS fixed x Can we do it ! ( ) ( ) Created by Mr. Lafferty@mathsrevision.com

  43. 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 ( ) ( ) Created by Mr. Lafferty@mathsrevision.com

  44. 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 ! ( ) ( ) Created by Mr. Lafferty@mathsrevision.com

  45. 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 ( ) ( ) Created by Mr. Lafferty@mathsrevision.com

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

  47. S3 Credit Starter Questions 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) Created by Mr. Lafferty@mathsrevision.com

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

  49. 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. Created by Mr. Lafferty@www.mathsrevision.com

  50. Summary of Factorising S3 Credit St. Andrew’s Cross method 2 squares Difference www.mathsrevision.com Take Out Common Factor Created by Mr. Lafferty@www.mathsrevision.com

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