Download Second Year Honours Maths Notes on Factors

Second Year Honours Maths Notes on Factors.
There are four types of factorising.
Factorising number one:

All the numbers have something in common.

Take out what they have in common and put everything else into brackets.
Example one:
x2 + 9x = x(x+9 )
by x to find out what goes into the bracket)
Example two:
x3 + x2 + x

x( x2 + x
=
+ 1)
There are four numbers and they don’t all have something
Factorising number two:
in common.




(divide each of the numbers
Pair them off.
Take out what they have in common and put everything else into brackets.
The brackets MUST be identical.
Open up two brackets back to back. Put the identical into one and the “leftovers”
into the other.
NOTE: If the only thing that is wrong with the “identical” brackets are the signs,
then change the signs inside and outside of one of the brackets.
5x2 + 10x y –
Example one: Factorise
x –
2y
5x2 – x
10 x y –
x(5x–1)
2 y ( 5 x - 1 ) (brackets are identical)
(5x–1)(x +2y )
( identical bracket ) ( leftover )
6x2 + 2 a –
Example two: Factorise
6x2 – 3ax
3x(2x
– a )
3x(2x–a)
( 2 x – a ) ( 3 x – 2 ).
3ax
2a –
2y
– 4x
4x
2(a- 2x)
- 2 ( -a + 2 x )
( signs wrong)
(now the brackets are identical)
Factorising number three:




The difference of two squares.
There can only be two numbers
There must be a minus between them.
They must be written as squares or can be written as squares.
Open up two brackets and add the numbers in the first, subtract the numbers in the
second.
Example one:
Factorise
x2 –
y2
(x+y)( x- y)
Example two:
Factorise
36a2 – 4 9y2
6 2a 2
-
72 y2
(6a+7y)( 6a- 7y ).
A difficult question!
Factorise 5 x 2 – 5 y 2
At first this looks like factorising number three but then you realise you cannot write 5 as
a square. Go back to the first type of factorising and they all have something in common
so take out the common factor and put everything else into the bracket.
5(x2 – y2 )
But you are not finished yet because you can now do factorising number three to it
5(x+y)(x- y)
Now you cannot do anything else to it so you are finished!
Factorising number four:
You have a quadratic expression, that means you have an x2 , an x and a number.
For example
x2 + 7x +1 2
 Open up two brackets
 Break up the x2 into x and x and put each into the beginning of each bracket.
 Then break up the end number into factors and put your answers into the end of
each bracket.
 If the second sign is + then both signs will be the same and if the second sign is –
then both signs will be different.


To make sure you have done this question correctly multiply the two outside
numbers together, multiply the two inside numbers together and add. You should
get back the middle number.
When there is a number in front of x2 the question becomes much harder so be
careful!
Example one:
x2 + 7 x + 12
(x +4) (x+ 3)
Example two:
x2 -2 x–3
(x–3)(x +1)
Example three:
36x2 – 7 x - 4
(4x+1)( 9x–4)