Factorising

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]

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