More complex algebraic expressions can often be factorised by extracting a common factor from all terms. This common factor may involve numbers, letters, powers or a combination of these.

 

Factorising in this way simplifies expressions and is an important step before solving equations or simplifying algebraic fractions.

 

 

Identifying the Common Factor

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

 

The common factor may include:
• a numerical factor
• a variable
• a power of a variable

 

For example, consider:

$$
12x^2y + 8xy
$$

Both terms share a factor of \( 4xy \).

 

 

Extracting a Common Factor

Once the common factor is identified, divide each term by it and place the result inside a bracket.

 

For example:

$$
12x^2y + 8xy
$$

 

Factor out \( 4xy \):

$$
4xy(3x + 2)
$$

 

Another example involving subtraction:

$$
15a^2 - 10a
$$

 

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

$$
5a(3a - 2)
$$

 

 

Factorising Expressions with Several Terms

Expressions with three or more terms can also be factorised if a common factor exists.

 

For example:

$$
6x^2 + 9x - 3
$$

 

All terms share a factor of \( 3 \):

$$
3(2x^2 + 3x - 1)
$$

 

Another example:

$$
10ab + 5a - 15a^2b
$$

 

The common factor is \( 5a \):

$$
5a(2b + 1 - 3ab)
$$

 

 
Including Negative Common Factors

Sometimes it is helpful to take out a negative common factor, especially if it makes the expression inside the bracket simpler.

 

For example:

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

 

Factor out \( -4 \):

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

 

This can make later steps, such as solving equations, clearer.

 
Key Points to Remember

Always look for the largest possible common factor.
The factor outside the bracket should multiply every term inside.
Factorising makes expressions easier to simplify and manipulate later.

 

Extracting common factors is a fundamental skill that underpins many other algebraic techniques.