Factorising means writing an expression as a product of its factors. If every term in an expression shares at least one common factor, this factor can be taken outside a bracket.

 

The common factor may be a number, a letter or a combination of both.

 

 
Identifying the Common Factor

To factorise, first identify the highest common factor of all terms. This is the largest factor that divides into every term.

 

For example, in the expression:

$$
6x + 9
$$

Both terms are divisible by \( 3 \), so \( 3 \) is the common factor.

 

 
Factorising Linear Expressions

Take the common factor outside the bracket.

 

For example:

$$
6x + 9
$$

 

Factor out \( 3 \):

$$
3(2x + 3)
$$

 

Another example:

$$
8a - 4
$$

 

Both terms share a factor of \( 4 \):

$$
4(2a - 1)
$$

 

If a variable is also common, include it in the factor.

$$
5x^2 + 10x
$$

 

Both terms share a factor of \( 5x \):

$$
5x(x + 2)
$$

 

 
Factorising Quadratic Expressions with a Common Factor

Quadratic expressions often have a common factor in all terms.

 

For example:

$$
3x^2 + 6x
$$

 

Both terms share \( 3x \):

$$
3x(x + 2)
$$

 

Another example:

$$
4x^2 - 8x + 12
$$

 

All terms share a factor of \( 4 \):

$$
4(x^2 - 2x + 3)
$$

 

Always check whether a larger common factor exists before factorising.

 

 
Important Points to Remember

Factorising is the reverse of expanding.
Always take out the largest common factor.
Every term inside the bracket should be simpler than before.

Factorising expressions correctly is essential for simplifying algebra, solving equations and working with algebraic fractions.