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Factorisation. Single Brackets. Multiply out the bracket below: 2x ( 4x – 6 ) = 8x 2 - 12x Factorisation is the reversal of the above process. That is to say we put the brackets back in. Example 1 Factorise: 4x 2 – 12 x = 4( x 2 – 3x) Hint:Numbers First. Hint:Now Letters = 4x( x – 3 ) Example 2 Factorise: 40x 2 – 5x =( 8x 2 - x )5 = 5 x( 8 x- 1 )

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What Goes In The Box ? Factorise fully : 12 x 2 – 6 x 6 ( 2x 2 - x ) 6x( 2x - 1 ) Now factorise the following: (1) 14 x 2 + 7 x (2) 4x – 12 x 2 (3) 6ab – 2ad (4) 12 a 2 b – 6 a b 2 =7x( x + 1) = 4x ( 1 – 3x) =2a( 3b – d) = 6ab ( a – b)

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A Difference Of Two Squares. Consider what happens when you multiply out : ( x + y ) ( x – y) = x( x – y )+ y( x – y ) =x 2 - xy+ xy - y 2 = x 2 - y 2 This is a difference of two squares. Now you try the example below: Example. Multiply out: ( 5 x + 7 y )( 5 x – 7 y ) Answer: = 25 x 2 - 49 y 2

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What Goes In The Box ? (1) ( 3 x + 6 y ) ( 3 x – 6 y) (2) ( 2 x – 4 y ) ( 2 x + 4 y) (3) ( 8 x + 9 y ) ( 8 x – 9 y) (3) ( 5 x – 7 y ) ( 5 x + 7 y) (4) ( x – 11 y ) ( x + 11 y) (5) ( 7 x + 2 y ) ( 7 x – 2 y) (6) ( 5 x – 9 y ) ( 5 x + 9 y) (7) ( 3 x + 9 y ) ( 3 x – 9 y) = 9 x 2 – 36 y 2 = 4 x 2 – 16 y 2 = 64 x 2 – 81 y 2 = 25 x 2 – 49 y 2 = x 2 – 121 y 2 = 49 x 2 – 4 y 2 = 25 x 2 – 81 y 2 = 9 x 2 – 81 y 2 Mutiply out:

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Factorising A Difference Of Two Squares. By considering the brackets required to produce the following factorise the following examples directly: Examples (1) x 2 - 9 (2) x 2 - 16 (3) x 2 - 25 (4) x 2 - y 2 (5) 4x 2 - 36 (6) 9x 2 - 16y 2 (7) 100g 2 - 49k 2 (8) 144d 2 - 36w 2 ( x - 3 )= ( x+ 3 ) = ( x - 4 )( x+ 4 ) = ( x - 5 )( x+ 5 ) = ( x - y )( x+ y ) = ( 2x - 6 )( 2x+ 6 ) = ( 3x - 4y )( 3x+ 4y ) = ( 10g – 7k )( 10g+ 7k ) = ( 12d - 6 w)( 12d+ 6w )

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What Goes In The Box ? Multiply out the brackets below: (3x – 4 ) ( 2x + 7) 3x(2x + 7)-4 (2x + 7) 6x 2 +21x -8x-28 6x 2 +13x -28 You are now about to discover how to put the double brackets back in.

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Factorising A Quadratic. Follow the steps below to put a double bracket back into a quadratic equation. Factorise the quadratic: x 2 – 2x - 15 Process. Step 1: Consider the factors of the coefficient in front of the x and the constant. Factors 115 111 35 Step 2 : Create the x coefficient from two pairs of factors. x coefficient = 2 (1 x 5) – (1 x 3 ) = 2 Step 3 Place the four numbers in the pair of brackets looking at outer and inner pairs to determine the signs. = (x 5) ( x 3) 5x 3x 3x – 5x = - 2x = (x - 5) ( x +3)

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More Quadratic Factorisation Examples. Example 1. Factorise the quadratic: x 2 + 3x - 10 Factors 110 111 25 x coefficient = 3 (1 x 5) - (1 x 2 ) = 3 = (x 5) ( x 2) 5x 2x Signs in brackets. = (x + 5) ( x - 2 ) 5x – 2x = 3x

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Quadratic Factorisation Example 2 Factorise the quadratic: x 2 – 8x + 12 Factors 112 111 3 6 x coefficient = 8 = (x 6) ( x 2) 6x 2x Signs in brackets. = (x - 6) ( x -2 ) - 6x – 2x = - 8x (1 x 6) + (1 x 2 ) = 8 2 4

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Quadratic Factorisation Example 3. Factorise the quadratic: 6 x 2 + 11x – 10 Factors 610 161 2325 x coefficient = 11 (3 x 5) – (2 x 2 ) = 11 = (3x 2) ( 2x 5) 4x 15x Signs in brackets. 15 x – 4x = 11x = ( 3x - 2) ( 2 x + 5) Numbers together. Numbers apart.

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Quadratic Factorisation Example 4 Factorise the quadratic: 10 x 2 + 27x – 28 Factors 1028 110128 25214 x coefficient = 27 (5 x 7) – (2 x 4 ) = 27 = (5x 4) ( 2x 7) 8x 35x Signs in brackets. 35 x – 8x = 27x = ( 5x - 4) ( 2 x + 7) 47

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What Goes In The Box ? Factorise the quadratic: 6 x 2 – x – 2 Factors 62 1612 23 x coefficient (2 x 2) – (1 x 3 ) = 1 = (3x 2) ( 2x 1) Signs in brackets. 3 x – 4x = -x = ( 3x - 2) ( 2 x + 1)

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What Goes In The Box 2 Factorise the quadratic: 15 x 2 – 19x + 6 Factors 156 1 16 35 x coefficient (3 x 3) + (5 x 2 ) = 19 = (3x 2) ( 5x 3) Signs in brackets. - 9 x – 10x = - 19x = ( 3x - 2) ( 5 x - 3) -19 23

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