Factorisation. Single Brackets.

1
Factorisation. Single Brackets.
Multiply out the bracket below 2x ( 4x 6 )
Example 2 Factorise 40x2 5x
8x2
- 12x

( 8x2
- x )
5
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
HintNumbers First.
4
( x2 3x)
HintNow 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
Factorising A Quadratic.
Follow the steps below to put a double bracket
back into a quadratic equation.
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)
6
2
3
2x
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
2
3
2
5
(3x 2) ( 2x 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