## What is Factorising?

Factorising is the reverse process of expanding brackets. A factorised answer will always contain a set of brackets. To factorise an expression fully, take out the highest common factor (HCF) of all the terms.

## How To Factorising

Here is the way hoe to factorising.

### Factorising – Expanding Brackets

This section shows you how to factorise and includes examples, sample questions, and videos.

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.

### General Form

If (x + a)(x + b) = x² + cx + d
= x² + (a+b)x + ab

where a + b = c and ab = d

Example

Expand (3x + 1)(x – 1)

Solution :

(3x + 1)(x – 1)
= 3x² – 3x + x – 1
2x² – 2x – 1

Factorising

Factorising is the reverse of expanding brackets, so it is, for example, putting 2x² -2x – 1 into the form (3x + 1)(x – 1). This is an important way of solving quadratic equations.
The first step of factorizing an expression is to ‘take out’ any common factors which the terms have. So if you were asked to factorize x² + 2x, since x goes into both terms, you would write x(x + 2).

There is no simple method of factorizing a quadratic expression, but with a little practice, 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 factorizing any expression easy. It is worth studying these examples further if you do not understand what is happening. Unfortunately, the only other method of factorizing is trial and error.

The Difference of Two Squares

If you are asked to factorize an expression which is one square number minus another, you can factorize it immediately. This is because a² – b² = (a + b)(a – b) .

Example :

Factorise 36 – x²
(6 + x)(6 – x)     [imagine that a = 6 and b = x]

Exercise :

Factorise the following :

• 49 – x²
• x² – 81

### Grouping

Grouping terms may allow using other methods for getting a factorization.

For example, to factor

one may remark that the first two terms have a common factor x, and the last two terms have the common factor of y. Thus

Then a simple inspection shows the common factor x + 5, leading to the factorization

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