Expanding Brackets
Brackets should be expanded in the following ways:
For an expression of
the form a(b + c), the expanded version is ab + ac, i.e., multiply the term
outside the bracket by everything inside the bracket (e.g. 2x(x + 3) = 2x² + 6x
[remember x × x is x²]).
For an expression of the form (a + b)(c + d), the
expanded version is ac + ad + bc + bd, in other words everything in the first
bracket should be multiplied by everything in the second.
Example
Expand (2x + 3)(x  1): (2x + 3)(x  1) = 2x²  2x + 3x
 3 = 2x² + x  3
Factorising
Factorising is the reverse of expanding brackets, so it is, for example,
putting 2x² + x  3 into the form (2x + 3)(x  1). This is an important way of
solving quadratic equations. The
first step of factorising an expression is to 'take out' any common factors
which the terms have. So if you were asked to factorise x² + x, since x goes
into both terms, you would write x(x + 1) .
Factorising Quadratics
There is no simple method of factorising a quadratic expression, but with a
little practise it becomes easier. One systematic method, however, is as
follows:
Example
Factorise 12y²  20y + 3 = 12y²  18y  2y + 3 [here
the 20y has been split up into two numbers whose multiple is 36. 36 was chosen
because this is the product of 12 and 3, the other two numbers]. The first
two terms, 12y² and 18y both divide by 6y, so 'take out' this factor of
6y. 6y(2y  3)  2y + 3 [we can do this because 6y(2y  3) is the same as
12y²  18y] Now, make the last two expressions look like the expression in
the bracket: 6y(2y  3) 1(2y  3) The answer is (2y  3)(6y 
1)
Example
Factorise x² + 2x  8 We need to split the 2x into two
numbers which multiply to give 8. This has to be 4 and 2. x² + 4x  2x 
8 x(x + 4)  2x  8 x(x + 4) 2(x + 4) (x + 4)(x 
2)
Once you work out what is going on, this method makes factorising
any expression easy. It is worth studying these examples further if you do not
understand what is happening. Unfortunately, the only other method of
factorising is by trial and error.
The Difference of Two Squares
If you are asked to factorise an expression which is one square number minus
another, you can factorise it immediately. This is because a²  b² = (a + b)(a 
b) .
Example
Factorise 25  x² = (5 + x)(5  x) [imagine
that a = 5 and b = x]
Copyright © Matthew Pinkney 2003
