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Factorisation. Single Brackets.Multiply out the bracket below:
2x ( 4x – 6 )
= 8x2 - 12x
Factorisation is the reversal of the above process. That is to say we put the brackets back in.
Example 1
Factorise: 4x2 – 12 x
= 4 ( x2 – 3x)
Hint:Numbers First.
Hint:Now Letters
= 4x ( x – 3 )
Example 2
Factorise: 40x2 – 5x
= ( 8x2 - x )5
= 5 x ( 8 x - 1 )
What Goes In The Box ?
Factorise fully :
12 x2 – 6 x
6 ( 2x2 - x )
6x ( 2x - 1 )
Now factorise the following:
(1) 14 x 2 + 7 x
(2) 4x – 12 x 2
(3) 6ab – 2ad
(4) 12 a2 b – 6 a b2
=7x( x + 1)
= 4x ( 1 – 3x)
=2a( 3b – d)
= 6ab ( a – b)
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
= x2 - y2
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
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:
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 )
= ( 10 g – 7k ) ( 10g + 7k )
= ( 12d - 6 w) ( 12d + 6w )
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.
Factorising A Quadratic.Follow the steps below to put a double bracket back into a quadratic equation.
Factorise the quadratic:
x2 – 2x - 15
Process. Step 1:
Consider the factors of the coefficient in front of the x and the constant.Factors
1 15
1 1 1 15
3 5
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)
More Quadratic Factorisation Examples.
Example 1.Factorise the quadratic:
x2 + 3x - 10
Factors
1 10
1 1 1 10
2 5
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
Quadratic Factorisation Example 2
Factorise the quadratic:
x2 – 8x + 12
Factors
1 12
1 1 1 12
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
Quadratic Factorisation Example 3.
Factorise the quadratic:
6 x2 + 11x – 10
Factors
6 10
1 6 1 10
2 3 2 5
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.
Quadratic Factorisation Example 4
Factorise the quadratic:
10 x2 + 27x – 28
Factors
10 28
1 10 1 28
2 5 2 14
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)
4 7
What Goes In The Box ?
Factorise the quadratic:
6 x2 – x – 2
Factors
6 2
1 6 1 2
2 3
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)
-1
What Goes In The Box 2Factorise the quadratic:
15 x2 – 19x + 6
Factors
15 6
1 15 1 6
3 5
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
2 3
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