Algebraic Operations Int 2 www mathsrevision com Removing

Algebraic Operations Int 2 www. mathsrevision. com Removing Brackets Pairs of Brackets Factors Common Factors Difference of Squares Factorising Trinomials (Quadratics) Factor Priority 23 -Oct-21 Created by Mr. Lafferty@mathsrevision. com

Starter Questions Int 2 www. mathsrevision. com Q 1. Calculate (a) -3 x 5 = (b) -6 x -7 = (b) -2 a Q 2. Calculate (a) wxw= x 4 a = Q 3. Find the gradient of the line if (3, 7) and (12, 34) 23 -Oct-21 Created by Mr. Lafferty@mathsrevision. com

Removing a Single Bracket www. mathsrevision. com Int 2 Learning Intention 1. To show to multiply out (remove) a single bracket. Success Criteria 1. Understand the keypoints of multiplying out a expression with a single bracket. 2. Be able multiply out a expression with a single bracket. 23 -Oct-21 Created by Mr. Lafferty@mathsrevision. com

Removing a Single Bracket Int 2 www. mathsrevision. com Example 1 3(b + 5) = 3 b + 15 Example 2 4(w - 2) = 4 w - 8 23 -Oct-21 Created by Mr. Lafferty@mathsrevision. com

Removing a Single Bracket Int 2 www. mathsrevision. com Example 3 a(y - 1) = ay - a Example 4 p(w - 6) = pw - 6 p 23 -Oct-21 Created by Mr. Lafferty@mathsrevision. com

Removing a Single Bracket Int 2 www. mathsrevision. com Example 5 x(x + 3) = x 2 + 3 x Example 6 3 q(3 q -2 m) = 9 q 2 - 6 mq 23 -Oct-21 Created by Mr. Lafferty@mathsrevision. com

Removing a Be careful with Single Bracket negatives !! Int 2 www. mathsrevision. com Example 7 -2(h + 5) = -2 h - 10 Example 8 -(g - 9) = -g + 9 23 -Oct-21 Created by Mr. Lafferty@mathsrevision. com

Removing a Single Bracket Int 2 www. mathsrevision. com Example 9 (x + 4) Find my Area 6(x + 4) = 6 x + 24 23 -Oct-21 Created by Mr. Lafferty@mathsrevision. com 6

Removing a Single Bracket Be careful only multiply everything inside the bracket www. mathsrevision. com Int 2 Example 10 8 +2(h + 3) = 8 + 2 h + 6 Now tidy up ! = 2 h + 14 23 -Oct-21 Created by Mr. Lafferty@mathsrevision. com

Removing a Single Bracket Be careful only multiply everything inside the bracket www. mathsrevision. com Int 2 Example 10 -2(y - 1) + 4 = -2 y + 2 + 4 Now tidy up ! = -2 y + 6 23 -Oct-21 Created by Mr. Lafferty@mathsrevision. com

Removing a Single Bracket Be careful only multiply everything inside the bracket www. mathsrevision. com Int 2 Example 11 y - (4 - y) = y - 4 + y Now tidy up ! =-4 23 -Oct-21 Created by Mr. Lafferty@mathsrevision. com

Removing a Single Bracket Int 2 www. mathsrevision. com Example 12 (x + 6) Find the area of the picture frame. x (x + 4) Area = x(x + 6) – 4(x + 4) 23 -Oct-21 Created by Mr. Lafferty@mathsrevision. com 4

Removing a Single Bracket Int 2 www. mathsrevision. com Example 12 Area = x(x + 6) – 4(x + 4) x 2 + 6 x - 4 x - 16 x 2 + 2 x - 16 23 -Oct-21 Created by Mr. Lafferty@mathsrevision. com Now tidy up !

Removing a Single Bracket Int 2 www. mathsrevision. com Example 13 x(x - 3) + 2(x - 3) x 2 - 3 x + 2 x - 6 x 2 - x - 6 23 -Oct-21 Created by Mr. Lafferty@mathsrevision. com Now tidy up !

Removing a Single Bracket www. mathsrevision. com Int 2 Now try Exercise 1 Ch 5 MIA (page 48) 23 -Oct-21 Created by Mr. Lafferty@mathsrevision. com

Starter Questions Int 2 www. mathsrevision. com Q 1. Calculate (a) -3 y 5 y = (b) -6 q x (-4 q) = a(b - c) = (b) -2 a( b – a) = x Q 2. Calculate (a) Q 3. Write down the gradient and were the line cuts the y – axis. y = 5 – 3 x 23 -Oct-21 Created by Mr. Lafferty@mathsrevision. com

Removing Double Brackets www. mathsrevision. com Int 2 Learning Intention 1. To show 2 methods for multiplying out brackets Success Criteria 1. Understand the keypoints of multiplying out double brackets. 2. Be able multiply out double brackets using 2 methods. 23 -Oct-21 Created by Mr. Lafferty@mathsrevision. com

Removing Double Brackets www. mathsrevision. com Int 2 There two methods we can use to multiply out DOUBLE brackets. First Method Simply remember the word FOIL Multiply First 2 Multiply Outside 2 23 -Oct-21 Created by Mr. Lafferty@mathsrevision. com Multiply Last 2 Multiply Inside 2

Removing Double Brackets www. mathsrevision. com Int 2 Example 1 : Multiply out the brackets and Simplify (x + 1)(x + 2) 1. Write down 2. Tidy up ! 23 -Oct-21 F O I L x 2 + 2 x + 2 x 2 + 3 x + 2 Created by Mr. Lafferty@mathsrevision. com

Removing a Single Bracket www. mathsrevision. com Int 2 Example 2 : Multiply out the brackets and Simplify (x - 1)(x + 2) 1. Write down 2. Tidy up ! 23 -Oct-21 F O I L x 2 + 2 x - 2 x 2 + x - 2 Created by Mr. Lafferty@mathsrevision. com

Removing Double Brackets www. mathsrevision. com Int 2 (x + 1)(x - 2) x 2 - x - 2 (x - 1)(x - 2) x 2 - 3 x + 2 (x + 3)(x + 2) x 2 + 5 x + 6 (x - 3)(x + 2) x 2 - x - 6 (x + 3)(x - 2) x 2 + x - 6 23 -Oct-21 Created by Mr. Lafferty@mathsrevision. com

Removing a Single Bracket www. mathsrevision. com Int 2 Now try Exercise 2 Q 1 Ch 5 MIA (page 50) 23 -Oct-21 Created by Mr. Lafferty@mathsrevision. com

Removing Double Brackets. We have Int 2 Another way is to make a Multiplication table Table www. mathsrevision. com (y y + 2)(y 2 y + 5) 5 23 -Oct-21 y 2 +5 y Tidy up ! +2 y +10 y 2 + 7 y +10 Created by Mr. Lafferty@mathsrevision. com

Removing Double Brackets. Be careful Int 2 with the negative signs Example 2 www. mathsrevision. com (2 x 2 x - 1)(x 1 x + 3) 3 Tidy up ! 2 x 2 +6 x -x 23 -Oct-21 -3 2 x 2 + 5 x - 3 Created by Mr. Lafferty@mathsrevision. com

Removing Double Brackets Just a bigger Int 2 Multiplication Table www. mathsrevision. com Example 3 (x 4)(x x 2 + 3 x + 2 2) x+4 x 3 +4 x 2 23 -Oct-21 Tidy up ! +3 x 2 +2 x +12 x +8 x 3 + 7 x 2 + 14 x + 8 Created by Mr. Lafferty@mathsrevision. com

Removing a Single Bracket www. mathsrevision. com Int 2 Now try Exercise 2 Q 3 – Q 5 Ch 5 MIA (page 51) 23 -Oct-21 Created by Mr. Lafferty@mathsrevision. com

Starter Questions Int 2 www. mathsrevision. com Q 1. Remove the brackets (a) a (4 y – 3 x) = (b) (2 x-1)(x+4) = Q 2. Calculate The interest on £ 20 over 5 years @ a compound interest of 7% per year. Q 3. Write down all the number that divide into 12 without leaving a remainder. 23 -Oct-21 Created by Mr. Lafferty@mathsrevision. com

Factors www. mathsrevision. com Int 2 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 23 -Oct-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 Int 2 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 23 -Oct-21 8 Created by Mr. Lafferty@mathsrevision. com

Factors www. mathsrevision. com Int 2 Highest Common Factor Largest Same Number We need to write out all factor pairs in order to find the Highest Common Factor. Created by Mr. Lafferty@mathsrevision. com

Factors www. mathsrevision. com Int 2 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 Created by Mr. Lafferty@mathsrevision. com

Factors www. mathsrevision. com Int 2 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 2 and 5 x , 5 and 2 x Created by Mr. Lafferty@mathsrevision. com 10 and x

Factors www. mathsrevision. com Int 2 Highest Common Factor Example : Find the HCF of ab and 2 b. F ab = 1 and ab Fx 2 = 1 and 2 b a and b Example : 2 and 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 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 Int 2 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 23 -Oct-21 Created by Mr. Lafferty@mathsrevision. com

Factors www. mathsrevision. com Int 2 Now try Exercise 3 Q 3 and Q 4 Ch 5 (page 52) 23 -Oct-21 Created by Mr. Lafferty

Starter Questions Int 2 www. mathsrevision. com Q 1. Remove the brackets (a) a (4 y – 3 x) = (b) (x + 5)(x - 5) = Q 2. For the line y = -x + 5, find the gradient and where it cuts the y axis. Q 3. Find the highest common factor for p 2 q and pq 2. 23 -Oct-21 Created by Mr. Lafferty@mathsrevision. com

Factorising www. mathsrevision. com Int 2 Using Factors Learning Intention 1. To show to factorise terms using the Highest Common Factor and one bracket term. 23 -Oct-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 Int 2 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 23 -Oct-21 15 ÷ 3 = 5 Created by Mr. Lafferty@www. mathsrevision. com 3 3( ) 3( x + 5 )

Check by multiplying Factorising Int 2 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 23 -Oct-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 Int 2 www. mathsrevision. com Factorise the following : (a) 3(x + 2) 3 x + 6 Be careful ! 2 x(y – 1) (b) 4 xy – 2 x (c) 6 a + 7 a 2 a(6 + 7 a) (d) y 2 - y y(y – 1) 23 -Oct-21 Created by Mr. Lafferty@mathsrevision. com

Factorising www. mathsrevision. com Int 2 Now try Exercise 4 Start at Q 2 Ch 5 (page 53) 23 -Oct-21 Created by Mr. Lafferty@www. mathsrevision. com

Starter Questions Int 2 www. mathsrevision. com Q 1. Remove the brackets (a) a (8 – 3 x + 6 a) = Q 2. Factorise 3 x 2 – 6 x Q 3. Write down the first 10 square numbers. 23 -Oct-21 Created by Mr. Lafferty@mathsrevision. com

Difference of Two Squares www. mathsrevision. com Int 2 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. 23 -Oct-21 Created by Mr. Lafferty@www. mathsrevision. com

Difference of Two Squares www. mathsrevision. com Int 2 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 23 -Oct-21 Difference Second square term Created by Mr. Lafferty@www. mathsrevision. com

Difference of Two Squares www. mathsrevision. com Int 2 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 23 -Oct-21 Created by Mr. Lafferty@www. mathsrevision. com

Difference of Two Squares www. mathsrevision. com Int 2 Keypoints Format a 2 – b 2 Always the difference sign ( a + b )( a – b ) 23 -Oct-21 Created by Mr. Lafferty

Difference of Two Squares Int 2 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 23 -Oct-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 Int 2 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 ) 23 -Oct-21 Created by Mr. Lafferty

Difference of Two Squares Int 2 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 23 -Oct-21 2( 2 + b )( 2 – b ) 3(3 w + 2 )( 3 w – 2 ) Created by Mr. Lafferty

Difference of Two Squares www. mathsrevision. com Int 2 Now try Exercise 5 Ch 5 (page 54) 23 -Oct-21 Created by Mr. Lafferty@www. mathsrevision. com

Starter Questions Int 2 www. mathsrevision. com Q 1. Multiple out the brackets and simplify. (a) ( y – 3 )( y + 6 ) Q 2. Factorise 49 – 4 x 2 Q 3. Write down an equation parallel to y = 4 x + 1 23 -Oct-21 Created by Mr. Lafferty@mathsrevision. com

Factorising Using FOIL www. mathsrevision. com Int 2 Learning Intention 1. To show to factorise trinomials ( quadratics) using FOIL. 23 -Oct-21 Success Criteria 1. Be able to factorise quadratics using FOIL. Created by Mr. Lafferty@www. mathsrevision. com

Factorising Using FOIL www. mathsrevision. com 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 23 -Oct-21 to factorise quadratics. Created by Mr. Lafferty@www. mathsrevision. com

Removing Double Brackets www. mathsrevision. com Int 2 A LITTLE REVISION Multiply out the brackets and Simplify (x + 1)(x + 2) 1. Write down 2. Tidy up ! 23 -Oct-21 F O I L x 2 + 2 x + 2 x 2 + 3 x + 2 Created by Mr. Lafferty@mathsrevision. com

Factorising Using FOIL Int 2 www. mathsrevision. com We can also use FOIL to go the opposite way (x + 1)(x + 2) 23 -Oct-21 FOIL Created by Mr. Lafferty@mathsrevision. com x 2 + 3 x+ 2

Factorising Using FOIL Int 2 Strategy for factorising quadratics www. mathsrevision. com x 2 F + 3 x+2 O+I L Put down two brackets ( x+1 )( x + 2 ) 23 -Oct-21 Created by Mr. Lafferty@mathsrevision. com F x x = x 2 O + 2 x + +x I + 3 x L 1 x 2 =+2 x

Factorising. O+I(-1)value 12 =will -12 be 1 (-12) (-1)x + 12 x==-12 11 x Using FOIL 6 = =-12 1 x(-2) + (-12 x) -11 x x x Int 2 x Sometimes it can be trick www. mathsrevision. com x 2 F + x - 12 O+I 2 x (-6) (-2 x) + 6 x= =-12 4 x x 4 = =-12 2 x(-3) + (-6 x) -4 x to get O+I correct 3 x (+4)4 x = -12 (-3 x) = -x 3 x + (-4) = -x x L Put down two brackets (x -3)( x + 4 ) 23 -Oct-21 ? Created by Mr. Lafferty@mathsrevision. com F x x = x 2 O + 4 x + -3 x I +x L -3 x 4 = -12

Factorising Using FOIL Int 2 www. mathsrevision. com Factorise using the difference of two squares (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 ) 23 -Oct-21 Created by Mr. Lafferty

Factorising Using FOIL www. mathsrevision. com Int 2 Now try Exercise 6 Ch 5 (page 56) 23 -Oct-21 Created by Mr. Lafferty@www. mathsrevision. com

Starter Questions www. mathsrevision. com Int 2 Q 1. Bacteria grows at a rate of 10% per hour. Initially there was 600 bacteria in dish. How many bacteria are there 5 hour later. Q 2. Find the volume of a cone with high 50 cm and diameter 10 cm Q 3. A line has gradient -7 and cuts the y axis at -5. Write down the equation of the line. 23 -Oct-21 Created by Mr. Lafferty@mathsrevision. com

Factorising Using FOIL www. mathsrevision. com Int 2 Learning Intention 1. To show to factorise trinomials ( quadratics) using FOIL. 23 -Oct-21 Success Criteria 1. Be able to factorise quadratics using FOIL. Created by Mr. Lafferty@www. mathsrevision. com

Factorising (-1)value 4 will = -4 be O+I Using FOIL 1 +(-4) 12 x (-1 x)==-4 11 x x Int 2 x Slightly harder example www. mathsrevision. com 3 x 2 - x - 4 F O+I (-12 x) -11 x (-4) +x 1 x 1 == -4 3 x 4 x = = -4 -x 4 x -(-1) -3 x + 4 x = x L Put down two brackets (3 x - 4 )( x + 1 ) 23 -Oct-21 Created by Mr. Lafferty@mathsrevision. com ? F 3 x x = 3 x 2 O + 3 x + -4 x I -x L -4 x 1 = -4 x

Factorising 8 x x = 8 x or Using FOIL 4 x 2 x = 8 x 2 x Int 2 x 2 www. mathsrevision. com Harder Still 8 x 2 F +22 x + 15 O+I L Put down two brackets ( 23 -Oct-21 )( ) F O + I L 15 x 1 = 15 or Created by Mr. Lafferty@mathsrevision. com 3 x 5 = 15 8 x 2 + 22 x +15

Factorising Using FOIL www. mathsrevision. com Int 2 We just have to try all combinations 8 x x x = 8 x 2 4 x x 2 x = 8 x 2 to see what works. O+I = +22 x Middle term (8 x + 1)(x+15) +121 x (4 x + 1)(2 x+15) +62 x (8 x + 15)(x+1) +23 x (4 x + 15)(2 x+1) +34 x (8 x + 3)(x+ 5) +29 x (4 x + 3)(2 x+ 5) +26 x (8 x + 5)(x+ 3) +43 x (4 x + 5)(2 x+ 3) +22 x 23 -Oct-21 Created by Mr. Lafferty@mathsrevision. com

Factorising Using FOIL Int 2 www. mathsrevision. com Factorise using the difference of two squares (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 ) 23 -Oct-21 Created by Mr. Lafferty

Factorising Using FOIL www. mathsrevision. com Int 2 Now try Exercise 7 Ch 5 (page 57) 23 -Oct-21 Created by Mr. Lafferty@www. mathsrevision. com

Starter Questions Int 2 www. mathsrevision. com Q 1. Multiple out the brackets and simplify. (a) ( 2 x – 5 )( x + 5 ) Q 2. Find the volume of a cylinder with high 6 m and diameter 9 cm Q 3. Find the gradient and where line cut y-axis. x=y+1 23 -Oct-21 Created by Mr. Lafferty@mathsrevision. com

Summary of Factorising www. mathsrevision. com Int 2 Learning Intention 1. To explain the factorising priorities. 23 -Oct-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 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. 23 -Oct-21 Factorise any quadratic expression left. Created by Mr. Lafferty@www. mathsrevision. com

If youof can successfully Summary complete this exercise then you have the Factorising necessary skills to pass Int 2 www. mathsrevision. com the algebraic part of the course. Now try Exercise 8 Ch 5 (page 57) 23 -Oct-21 Created by Mr. Lafferty@www. mathsrevision. com