Factorisation of an algebraic expression is the reverse process of expanding.
It involves placing brackets into an expression.
A sum of terms then becomes a product.
Once an expression has been factorised, it should then be expanded to check it is correct.
Some expressions cannot be factorised.
If every term of the expression has a common factor, this common factor should be removed and placed outside a set of brackets. Always look for a common factor first.
e.g. 8x + 16 = 8(x + 2) the common factor is 8
6xy − 18x = 6x(y − 3) the common factor is 6x
The highest or largest common factor must be taken out. e.g. If you only take out a 6 from the example above the expression will not be fully factorised.
Sometimes terms need to be grouped together to find common factors.
e.g. ab + 2b + 5a + 10
= b(a + 2) + 5(a + 2) the common factor is (a + 2)
= (a + 2)(b + 5)
The aim when factorising quadratics should be to be able to do them mentally.
A trial-and-error process is used, then checked by expanding.
There are basically two types of quadratics:
1. If the coefficient of x2 is 1.
e.g. Factorise x2 + 6x + 8
- The x2 term must come from an x in each bracket. (x + ... )(x + ....)
- The + 8 must come from two numbers multiplied together.
+8 multiplied by +1OR
+4 multiplied by +2OR
-8 multiplied by -1OR
-4 multiplied by -2
- The + 6 must result from adding one of the above pairs of numbers.
The only correct two numbers would be + 4 and + 2.
Therefore x2+ 6x + 8 = (x + 4)(x + 2)
Click here to practice factorising quadratics.
2. If the coefficient of x2 is not 1.
A trial-and-error process is again used, with checking done by expanding.
Difference of Two Squares
a2 − b2 = (a + b)(a − b)
This property helps to factorise expressions containing two squared terms that are being subtracted.
|x2 − 9||= (x + 3)(x − 3)|
|81q2 − 100p2||= (9q + 10p)(9q − 10p)|
The patterns of the perfect squares of the previous section on expanding, should be learned to assist with factorisation.
e.g. a2 + 2ab + b2 = (a + b)2
a2 − 2ab + b2 =(a − b)2
|x2 − 4x + 4||= (x − 2)(x − 2) = (x − 2)2|
|4x4 + 12x2 + 9||= (2x2 + 3)(2x2 + 3)|