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