Transcript Factorisation. Single Brackets.

Factorisation. Single Brackets.
Multiply out the bracket below:
2x ( 4x – 6 )
Factorise: 40x2 – 5x
= 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 Hint:Numbers First.
= 4 ( x2 – 3x)
= 4x ( x – 3 )
Example 2
Hint:Now Letters
= 5 ( 8x2 - x )
=5 x( 8 x - 1 )
What Goes In The Box ?
Factorise fully :
12 x2 – 6 x
6
( 2x2
-x)
Now factorise the following:
(1) 14 x 2 + 7 x
=7x( x + 1)
(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)
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 ?
Mutiply out:
(1) ( 3 x + 6 y ) ( 3 x – 6 y)
= 9 x 2 – 36 y 2
(4) ( x – 11 y ) ( x + 11 y)
(2) ( 2 x – 4 y ) ( 2 x + 4 y)
= x 2 – 121 y 2
(5) ( 7 x + 2 y ) ( 7 x – 2 y)
= 4 x 2 – 16 y 2
= 49 x 2 – 4 y 2
(3) ( 8 x + 9 y ) ( 8 x – 9 y)
= 64 x 2 – 81 y 2
= 25 x
(3) ( 5 x – 7 y ) ( 5 x + 7 y)
= 25 x
2
– 49 y
(6) ( 5 x – 9 y ) ( 5 x + 9 y)
2
2
– 81 y 2
(7) ( 3 x + 9 y ) ( 3 x – 9 y)
=9x2
– 81 y 2
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
(5) 4x 2 - 36
=(x -3) (x +3)
(2) x 2 - 16
=(x -4) (x +4)
(3) x 2 - 25
=(x -5)
2
(4) x - y
=(x -y)
(x +5)
2
(x +y)
= ( 2x - 6 ) ( 2x + 6 )
(6) 9x 2 - 16y 2
= ( 3x - 4y ) ( 3x + 4y )
(7) 100g 2 - 49k 2
= ( 10 g – 7k ) ( 10g + 7k )
(8) 144d 2 - 36w 2
= ( 12d - 6 w) ( 12d + 6w )
What Goes In The Box ?
Multiply out the brackets below:
(3x – 4 ) ( 2x + 7)
3x
(2x + 7)
6x 2
+21x
6x
2
+13x
-4
-8x
-28
(2x + 7)
-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.
Process.
Factorise the quadratic:
x2 – 2x - 15
Consider the factors of the
coefficient in front of the x and
the constant.
Factors
5x
1
15
= (x 5) ( x 3)
3x
3x – 5x = - 2x
= (x - 5) ( x +3)
1
1
Step 1:
1
15
3
5
Step 2 :
Create the x coefficient from
two pairs of factors.
Step 3
x coefficient = 2
(1 x 5) – (1 x 3 ) = 2
Place the four numbers in the
pair of brackets looking at
outer and inner pairs to
determine the signs.
More Quadratic Factorisation Examples.
Example 1.
Factorise the quadratic:
Factors
1
x2 + 3x - 10
5x
= (x 5) ( x 2)
2x
= (x + 5) ( x - 2 )
1
10
1
1
10
2
5
x coefficient = 3
(1 x 5) - (1 x 2 ) = 3
Signs in brackets.
5x – 2x = 3x
Quadratic Factorisation Example 2
Factorise the quadratic:
Factors
1
x2 – 8x + 12
6x
= (x 6) ( x 2)
2x
= (x - 6) ( x -2 )
1
12
1
1
12
2
3
6
4
x coefficient = 8
(1 x 6) + (1 x 2 ) = 8
Signs in brackets.
- 6x – 2x = - 8x
Quadratic Factorisation Example 3.
Factorise the quadratic:
Factors
6
6 x2 + 11x – 10
4x
= (3x 2) ( 2x 5)
15x
= ( 3x - 2) ( 2 x + 5)
Numbers together.
Numbers apart.
10
1
6
1
10
2
3
2
5
x coefficient = 11
(3 x 5) – (2 x 2 ) = 11
Signs in brackets.
15 x – 4x = 11x
Quadratic Factorisation Example 4
Factorise the quadratic:
Factors
10
10 x2 + 27x – 28
8x
= (5x 4) ( 2x 7)
35x
= ( 5x - 4) ( 2 x + 7)
28
1
10
1
28
2
5
2
14
4
7
x coefficient = 27
(5 x 7) – (2 x 4 ) = 27
Signs in brackets.
35 x – 8x = 27x
What Goes In The Box ?
Factorise the quadratic:
Factors
6
6 x2 – x – 2
= (3x 2) ( 2x 1)
2
1
6
2
3
1
x coefficient
= ( 3x - 2) ( 2 x + 1)
2
-1
(2 x 2) – (1 x 3 ) = 1
Signs in brackets.
3 x – 4x = -x
What Goes In The Box 2
Factorise the quadratic:
15
x2 –
Factors
15
19x + 6
= (3x 2) ( 5x 3)
6
1
15
1
3
5
2
x coefficient
= ( 3x - 2) ( 5 x - 3)
6
3
-19
(3 x 3) + (5 x 2 ) = 19
Signs in brackets.
- 9 x – 10x = - 19x