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Factorisation. Single Brackets.

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

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Factorisation. Single Brackets.

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  1. Factorisation. Single Brackets. Multiply out the bracket below: 2x ( 4x – 6 ) Example 2 Factorise: 40x2 – 5x = 8x2 - 12x = 5 ( 8x2 - x ) Factorisation is the reversal of the above process. That is to say we put the brackets back in. = 5 x ( 8 x - 1 ) Example 1 Factorise: 4x2 – 12 x Hint:Numbers First. = 4 ( x2 – 3x) Hint:Now Letters = 4x ( x – 3 )

  2. What Goes In The Box ? Factorise fully : 12 x2 – 6 x Now factorise the following: (1) 14 x 2 + 7 x =7x( x + 1) 6 ( 2x2 - x ) (2) 4x – 12 x 2 = 4x ( 1 – 3x) 6x ( 2x - 1 ) (3) 6ab – 2ad =2a( 3b – d) (4) 12 a2 b – 6 a b2 = 6ab ( a – b)

  3. A Difference Of Two Squares. Consider what happens when you multiply out : ( x + y ) ( x – y) Now you try the example below: Example. Multiply out: ( 5 x + 7 y )( 5 x – 7 y ) = x ( x – y ) + y ( x – y ) - y 2 =x 2 - xy + xy Answer: = x2 - y2 = 25 x 2 - 49 y 2 This is a difference of two squares.

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

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

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

  7. Follow the steps below to put a double bracket back into a quadratic equation. Factorising A Quadratic. Process. Step 1: Consider the factors of the coefficient in front of the x and the constant. Factorise the quadratic: x2 – 2x - 15 Factors 5x 1 15 Step 2 : Create the x coefficient from two pairs of factors. = (x 5) ( x 3) 1 1 1 15 3x 3 5 Step 3 Place the four numbers in the pair of brackets looking at outer and inner pairs to determine the signs. 3x – 5x = - 2x x coefficient = 2 (1 x 5) – (1 x 3 ) = 2 = (x - 5) ( x +3)

  8. More Quadratic Factorisation Examples. Example 1. Factorise the quadratic: x2 + 3x - 10 Factors 1 10 1 1 1 10 5x = (x 5) ( x 2) 2 5 2x x coefficient = 3 (1 x 5) - (1 x 2 ) = 3 = (x + 5) ( x - 2 ) Signs in brackets. 5x – 2x = 3x

  9. Quadratic Factorisation Example 2 Factorise the quadratic: x2 – 8x + 12 Factors 1 12 1 1 1 12 6x = (x 6) ( x 2) 2 6 2x 3 4 x coefficient = 8 = (x - 6) ( x -2 ) (1 x 6) + (1 x 2 ) = 8 Signs in brackets. - 6x – 2x = - 8x

  10. Quadratic Factorisation Example 3. Factorise the quadratic: 6 x2 + 11x – 10 Factors 6 10 1 6 1 10 4x = (3x 2) ( 2x 5) 2 3 2 5 15x x coefficient = 11 = ( 3x - 2) ( 2 x + 5) (3 x 5) – (2 x 2 ) = 11 Numbers together. Signs in brackets. Numbers apart. 15 x – 4x = 11x

  11. Quadratic Factorisation Example 4 Factorise the quadratic: 10 x2 + 27x – 28 Factors 10 28 1 10 1 28 8x 2 5 2 14 = (5x 4) ( 2x 7) 4 7 35x x coefficient = 27 = ( 5x - 4) ( 2 x + 7) (5 x 7) – (2 x 4 ) = 27 Signs in brackets. 35 x – 8x = 27x

  12. What Goes In The Box ? Factorise the quadratic: 6 x2 – x – 2 Factors 6 2 1 6 1 2 2 3 = (3x 2) ( 2x 1) x coefficient -1 = ( 3x - 2) ( 2 x + 1) (2 x 2) – (1 x 3 ) = 1 Signs in brackets. 3 x – 4x = -x

  13. What Goes In The Box 2 Factorise the quadratic: 15 x2 – 19x + 6 Factors 15 6 1 15 1 6 3 5 2 3 = (3x 2) ( 5x 3) x coefficient -19 = ( 3x - 2) ( 5 x - 3) (3 x 3) + (5 x 2 ) = 19 Signs in brackets. - 9 x – 10x = - 19x

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