Int 2 Algebraic Operations Removing Brackets Difference of Squares Pairs of Brackets Factors Common Factors Factorising Trinomials (Quadratics) Factor Priority 2-May-15

Starter Questions Q1.Calculate (a)-3 x 5 =(b)-6 x -7 = Q2.Calculate (a)w x w =(b)-2a x 4a = Int 2 Q3.Find the gradient of the line if (3, 7) and (12, 34)

2-May-15 Learning Intention Success Criteria 1.To show how to multiply out (remove) a single bracket. 1.Understand the keypoints of multiplying out a expression with a single bracket. Int 2 2.Be able multiply out a expression with a single bracket. Removing a Single Bracket

Int 2 3(b + 5) =3b + 15 Example 1 4(w - 2) =4w - 8 Example 2 2-May-15 Removing a Single Bracket

Int 2 a(y - 1) =ay - a Example 3 p(w - 6) =pw - 6p Example 4 2-May-15 Removing a Single Bracket

Int 2 x(x + 3) =x2x2 + 3x Example 5 3q(3q -2m) =9q 2 - 6mq Example 6 2-May-15 Removing a Single Bracket

Int 2 -2(h + 5) =-2h - 10 Example 7 -(g - 9) =-g + 9 Example 8 2-May-15 Removing a Single Bracket Be careful with negatives !!

Int 2 6(x + 4) = 6x + 24 Example 9 2-May-15 Removing a Single Bracket Find my Area (x + 4) 6

Int (h + 3) =8 + 6 Example 10 2-May-15 Removing a Single Bracket Be careful only multiply everything inside the bracket + 2h Now tidy up ! + 14= 2h

Int 2 -2(y - 1) + 4 =-2y + 4 Example 10 2-May-15 Removing a Single Bracket Be careful only multiply everything inside the bracket + 2 Now tidy up ! + 6= -2y

Int 2 y - (4 - y) =y + y Example 11 2-May-15 Removing a Single Bracket Be careful only multiply everything inside the bracket - 4 = - 4 Now tidy up !

Int 2 x(x + 6) Example 12 Find the area of the picture frame. 2-May-15 Removing a Single Bracket (x + 6) 4 x (x + 4) Area = 4(x + 4)–

Int 2 x(x + 6) – 4(x + 4) x2x2 + 6x Example 12 2-May-15 Removing a Single Bracket Area = - 4x- 16 x2x2 + 2x- 16 Now tidy up !

Int 2 x(x - 3) + 2(x - 3) x2x2 - 3x Example 13 2-May-15 Removing a Single Bracket + 2x- 6 x2x2 - x- 6 Now tidy up !

2-May-15 Now try Exercise 1 Ch5 MIA (page 48) Int 2 Removing a Single Bracket

2-May-15 Starter Questions Q1.Calculate (a)-3y x 5y =(b)-6q x (-4q) = Q2.Calculate (a)a(b - c) =(b)-2a( b – a) = Int 2 Q3.Write down the gradient and were the line cuts the y – axis.y = 5 – 3x

2-May-15 Learning Intention Success Criteria 1.To show 2 methods for multiplying out brackets 1.Understand the keypoints of multiplying out double brackets. Int 2 2.Be able multiply out double brackets using 2 methods. Removing Double Brackets

Int 2 There two methods we can use to multiply out DOUBLE brackets. 2-May-15 Removing Double Brackets Simply remember the word FOIL Multiply First 2 Multiply Last 2 Multiply Outside 2 Multiply Inside 2 First Method

Int 2 (x + 1)(x + 2) x2x2 + 2x Example 1 : Multiply out the brackets and Simplify 2-May-15 Created by Mr. 1.Write down F O I L + x Tidy up ! Removing Double Brackets

Int 2 (x - 1)(x + 2) x2x2 + 2x Example 2 : Multiply out the brackets and Simplify 2-May-15 Created by Mr. Removing a Single Bracket 1.Write down F O I L - x Tidy up !

Int 2 2-May-15 (x + 1)(x - 2) Removing Double Brackets (x - 1)(x - 2) (x + 3)(x + 2) (x - 3)(x + 2) (x + 3)(x - 2) x 2 - x - 2 x 2 - 3x + 2 x 2 + 5x + 6 x 2 - x - 6 x 2 + x - 6

2-May-15 Now try Exercise 2 Q1 Ch5 MIA (page 50) Int 2 Removing a Single Bracket

Int 2 “the wee table method” 2-May-15 Removing Double Brackets (y + 2)(y + 5)y+ 2y+ 5 We have Multiplication Table +5y +10+2y y 2 Tidy up ! y 2 + 7y +10

Int 2 Example 2 2-May-15 Removing Double Brackets (2x - 1)(x + 3)2x- 1x+ 3 Be careful with the negative signs +6x -3 -x 2x 2 Tidy up ! 2x 2 + 5x - 3

Int 2 Example 3 2-May-15 Removing Double Brackets (x + 4)(x 2 + 3x + 2) x+ 4 x2x2 + 3x Just a bigger Multiplication Table +3x 2 +12x +4x 2 x 3 Tidy up ! x 3 + 7x x x +8

2-May-15 Now try Exercise 2 Ch5 MIA (page 50) Int 2 Removing a Single Bracket

2-May-15 Starter Questions Q1.Remove the brackets (a)a (4y – 3x) =(b)(2x-1)(x+4) = Q2.Calculate The interest on £20 over 5 a compound interest of 7% per year. Int 2 Q3.Write down all the number that divide into 12 without leaving a remainder.

2-May-15 Learning Intention Success Criteria 1.To identify factors using factor pairs 1.To explain that a factor divides into a number without leaving a remainder 2.To explain how to find Highest Common Factors 2.Find HCF for two numbers by comparing factors. Factors Using Factors Int 2

2-May-15 Factors Factors Example :Find the factors of 56. F56 =1 and 56 Always divide by 1 and find its pair 2 and 28 4 and 14 7 and 8 From 2 find other factors and their pairs Int 2

2-May-15 Factors Highest Common Factor We need to write out all factor pairs in order to find the Highest Common Factor. Highest Common Factor Largest Same Number Int 2

F8 =1 and May-15 Example :Find the HCF of 8 and 12. HCF = 4 F12 = 1 and 12 2 and Highest Common Factor Factors Int 2

F4x =1, and 4x, 2 and 2x 4 and x 2-May-15 Example :Find the HCF of 4x and x 2. HCF = x Fx 2 = 1 and x2x2 x x Highest Common Factor F5 = 1 and 5 Example :Find the HCF of 5 and 10x. HCF = 5 F10x = 1, and 10x 2 and 5x, 5 and 2x 10 and x Factors Int 2

F ab =1 and ab a and b 2-May-15 Example :Find the HCF of ab and 2b. HCF = b Fx 2 = 1 and 2b 2 and b Highest Common Factor F 2h 2 = 1 and 2h 2 2 and h2 h2, h 2h Example :Find the HCF of 2h 2 and 4h. HCF = 2h F4h = 1 and 4h 2 and 2h 4 and h Factors Int 2

2-May-15 Factors Find the HCF for these terms (a)16w and 24w (b) 9y 2 and 6y (c) 4h and 12h 2 (d)ab 2 and a 2 b 8w 3y 4h ab Int 2

2-May-15 Now try Exercise 3 Q3 and Q4 Ch5 (page 52) Factors Int 2

2-May-15 Starter Questions Q1.Remove the brackets (a)a (4y – 3x) =(b)(x + 5)(x - 5) = Q2.For the line y = -x + 5, find the gradient and where it cuts the y axis. Int 2 Q3.Find the highest common factor for p 2 q and pq 2.

2-May-15 Learning Intention Success Criteria 1.To identify the HCF for given terms. 1.To show how to factorise terms using the Highest Common Factor and one bracket term. 2.Factorise terms using the HCF and one bracket term. Factorising Using Factors Int 2

2-May-15 Factorising Example Factorise 3x Find the HCF for 3x and HCF goes outside the bracket3( ) 3.To see what goes inside the bracket divide each term by HCF 3x ÷ 3 = x15 ÷ 3 = 53( x + 5 ) Check by multiplying out the bracket to get back to where you started Int 2

2-May-15 Factorising Example 1.Find the HCF for 4x 2 and 6xy2x 2.HCF goes outside the bracket2x( ) 3.To see what goes inside the bracket divide each term by HCF 4x 2 ÷ 2x =2x6xy ÷ 2x = 3y2x( 2x- 3y ) Factorise 4x 2 – 6xy Check by multiplying out the bracket to get back to where you started Int 2

2-May-15 Factorising Factorise the following : (a)3x + 6 (b) 4xy – 2x (c) 6a + 7a 2 (d)y 2 - y 3(x + 2) 2x(y – 1) a(6 + 7a) y(y – 1) Be careful ! Int 2

2-May-15 Now try Exercise 4 Start at Q2 Ch5 (page 53) Factorising Int 2

2-May-15 Starter Questions Q1.Remove the brackets (a)a (8 – 3x + 6a) = Q2.Factorise 3x 2 – 6x Int 2 Q3.Write down the first 10 square numbers.

2-May-15 Learning Intention Success Criteria 1.Recognise when we have a difference of two squares. 1.To show how to factorise the special case of the difference of two squares. 2.Factorise the difference of two squares. Difference of Two Squares Int 2

2-May-15 When we have the special case that 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 Second square term Difference Difference of Two Squares Int 2

2-May-15 a 2 – b 2 First square term Second square term Difference This factorises to ( a + b )( a – b ) Two brackets the same except for + and a - Check by multiplying out the bracket to get back to where you started Difference of Two Squares Int 2

2-May-15 Keypoints Formata 2 – b 2 Always the difference sign - ( a + b )( a – b ) Difference of Two Squares Int 2

2-May-15 Factorise using the difference of two squares (a)x 2 – y 2 (b) w 2 – z 2 (c) 9a 2 – b 2 (d)16y 2 – 100k 2 (x + y )( x – y ) ( w + z )( w – z ) ( 3a + b )( 3a – b ) ( 4y + 10k )( 4y – 10k ) Difference of Two Squares Int 2

2-May-15 Trickier type of questions to factorise. Sometimes we need to take out a common And the use the difference of two squares. ExampleFactorise2a ( a + 3 )( a – 3 ) Difference of Two Squares Int 2 First take out common factor 2(a 2 - 9) Now apply the difference of two squares

2-May-15 Factorise these trickier expressions. (a)6x 2 – 24 (b) 3w 2 – 3 (c) 8 – 2b 2 (d) 27w 2 – 12 6(x + 2 )( x – 2 ) 3( w + 1 )( w – 1 ) 2( 2 + b )( 2 – b ) 3(3 w + 2 )( 3w – 2 ) Difference of Two Squares Int 2

2-May-15 Now try Exercise 5 Ch5 (page 54) Difference of Two Squares Int 2

2-May-15 Starter Questions Q1.Multiple out the brackets and simplify. (a)( y – 3 )( y + 6 ) Q2.Factorise 49 – 4x 2 Int 2 Q3.Write down an equation parallel to y = 4x + 1

2-May-15 Learning Intention Success Criteria 1.Be able to factorise quadratics using FOIL. 1.To show how to factorise trinomials ( quadratics) using FOIL. Int 2 Factorising Using FOIL

2-May-15 Factorising Using FOIL Int 2 There various ways of factorising trinomials ( quadratics) e.g. The ABC method, St. Andrew’s cross method. We will use our previous knowledge and use the FOIL METHOD to factorise quadratics.

Int 2 (x + 1)(x + 2) x2x2 + 2x A LITTLE REVISION Multiply out the brackets and Simplify 2-May-15 1.Write down F O I L + x Tidy up ! x 2 + 3x + 2 Removing Double Brackets

Int 2 (x + 1)(x + 2)x2x2 + 3x We can also use FOIL to go the opposite way 2-May FOIL (x + 1)(x + 2) x2x2 + 3x + 2 FOIL Factorising Using FOIL

Int 2 + 3x ( )( ) + 2x 2-May-15 Put down two brackets + x x 2 + 3x+2 Strategy for factorising quadratics x 2 +2 x x x = 1 x 2 = xx+1+ 2 F O+I L Factorising Using FOIL

Int 2 Sometimes it can be trick to get O+I correct + x ( )( ) + 4x 2-May-15 Put down two brackets -3 x x 2 + x - 12 x x x x = -3 x 4 = xx-3+ 4 F O+I L Factorising Using FOIL O+I value will be (-1)x + 12x = 11x 1x + (-12x) = -11x (-2x) + 6x = 4x 2x + (-6x) = -4x (-3x) + 4x = +x 3x + (-4) = -x (-1) x 12 = x (-12) = -12 (-2) x 6 = x (-6) = -12 (-3) x 4 = x ( 4) = -12 ?

2-May-15 Factorise using the difference of two squares (a)m 2 + 2m +1 (b) y 2 + 6m + 5 (c) b 2 – b -2 (d)a 2 – 5a + 6 (m + 1 )( m + 1 ) ( y + 5 )( y + 1 ) ( b - 2 )( b + 1 ) ( a - 3 )( a – 2 ) Int 2 Factorising Using FOIL

2-May-15 Now try Exercise 6 Ch5 (page 56) Factorising Using FOIL Int 2

2-May-15 Starter Questions Q1.Bacteria grows at a rate of 10% per hour. Initially there was 600 bacteria in dish. How many bacteria are there 5 hour later. Q2.Find the volume of a cone with high 50cm and diameter 10cm Int 2 Q3.A line has gradient -7 and cuts the y axis at -5. Write down the equation of the line.

2-May-15 Learning Intention Success Criteria 1.Be able to factorise quadratics using FOIL. 1.To show how to factorise trinomials ( quadratics) using FOIL. Int 2 Factorising Using FOIL

Int 2 Factorising Using FOIL 3x x x =3x 2 ( )( ) - 4 (-1) x 4 = -4 1 x (-4) = -4 (-4) x 1 = -4 4 x (-1) = -4 ? Slightly harder example - x + 3x 2-May-15 Put down two brackets - 4 x 3x 2 - x x 1 = 3xx+ 1 F O+I L O+I value will be 12x + (-1x) = 11x (-12x) + 1x = -11x 3x - 4x = -x -3x + 4x = x

Int 2 Harder Still + 22x ( )( ) 2-May-15 Put down two brackets 8x x x FO+I L Factorising Using FOIL 8x x x = 8x 2 or 4x x 2x = 8x 2 15 x 1 = 15 or 3 x 5 = 15

Int 2 We just have to try all combinations to see what works. 2-May x Factorising Using FOIL 8x x x = 8x 2 (8x + 1)(x+15) (8x + 15)(x+1) (8x + 3)(x+ 5) (8x + 5)(x+ 3) 4x x 2x = 8x 2 (4x + 1)(2x+15) (4x + 15)(2x+1) (4x + 3)(2x+ 5) (4x + 5)(2x+ 3) +23x +29x +43x +62x +34x +26x +22x Middle term O+I = +22x

2-May-15 Factorise using the difference of two squares (a)m 2 + 2m +1 (b) y 2 + 6m + 5 (c) b 2 – b - 2 (d)a 2 – 5a + 6 (m + 1 )( m + 1 ) ( y + 5 )( y + 1 ) ( b - 2 )( b + 1 ) ( a - 3 )( a – 2 ) Int 2 Factorising Using FOIL

2-May-15 Now try Exercise 7 Ch5 (page 57) Factorising Using FOIL Int 2

2-May-15 Starter Questions Q1.Multiple out the brackets and simplify. (a)( 2x – 5 )( x + 5 ) Int 2 Q3.Find the gradient and where line cut y-axis. x = y + 1 Q2.Find the volume of a cylinder with high 6m and diameter 9cm

2-May-15 Learning Intention Success Criteria 1.Be able use the factorise priorities to factorise various expressions. 1.To explain the factorising priorities. Int 2 Summary of Factorising

2-May-15 Summary of Factorising Int 2 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.Factorise any quadratic expression left.

2-May-15 Now try Exercise 8 Ch5 (page 57) Int 2 Summary of Factorising If you can successfully complete this exercise then you have the necessary skills to pass the algebraic part of the course.