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.

Common Factors |
Grouping |
Quadratics |
Difference of Two Squares |
Perfect Squares |

Common Factors

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.

### Grouping

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)

### Quadratics

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 x ^{2} is 1.**

e.g. Factorise x^{2} + 6x + 8

- The x
^{2}term must come from an x in each bracket. (x + ... )(x + ....) - The + 8 must come from two numbers multiplied together.

+8 multiplied by +1

**OR**+4 multiplied by +2

**OR**-8 multiplied by -1

**OR**-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 x ^{2}+ 6x + 8 = (x + 4)(x + 2)**

Click here to practice factorising quadratics.

**2. If the coefficient of x ^{2} is not 1.**

A trial-and-error process is again used, with checking done by expanding.

e.g. 2x^{2}+ 13x + 6 = (2x + 1)(x + 6)

- The 2x and the x are multiplied to give 2x
^{2} - The + 6 and the + 1 are multiplied to give + 6.
- The middle term comes from 2x . 6 and 1 . x which add to give 13x.

**Practice makes Perfect!**

### Difference of Two Squares

a^{2} − b^{2} = (a + b)(a − b)

This property helps to factorise expressions containing two squared terms that are being subtracted.

Example |
Factorised |

x^{2} − 9 |
= (x + 3)(x − 3) |

81q^{2 −} 100p^{2} |
= (9q + 10p)(9q − 10p) |

### Perfect Squares

The patterns of the perfect squares of the previous section on expanding, should be learned to assist with factorisation.

e.g. a^{2} + 2ab + b^{2} = (a + b)^{2}

a^{2} − 2ab + b^{2} =(a − b)^{2}

Example |
Factorised |

x^{2} − 4x + 4 |
= (x − 2)(x − 2) = (x − 2)^{2} |

4x^{4} + 12x^{2} + 9 |
= (2x^{2} + 3)(2x^{2} + 3) |