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We have seen that the distributive law can be used to rewrite a product involving brackets as an expression without brackets.

For instance, the product \(3(a+2)\) can be rewritten as \(3a+6\); this is called the expanded form of the expression and the process is called expansion.

The process of writing an algebraic expression as a product of two or more algebraic factors is called factorisation. Factorisation is the reverse process to expansion. In the example above, \(3(a+2)\) is the factorised form of the algebraic expression.

Factorise using common factors

If each term in the algebraic expression to be factorised contains a common factor, then this common factor is a factor of the entire expression. To find the other factor, we divide each term by the common factor. The common factor is placed outside brackets. For this reason, the process is sometimes called 'taking the common factor outside the brackets'.

Example 1

Factorise \(4a+12\).


4 is a common factor of \(4a\) and 12 thus


Example 2

Factorise \(3x+9\).


3 is a common factor of \(3x\) and 9 thus


Example 3

Factorise \(x^2-4x\).


\(x\) is a common factor of \(x^2\) and \(4x\) thus


Example 4

Factorise \(2xy+4x^2-6xz\).


\(2x\) is a common factor of \(2xy, 4x^2\ \text{and}\ 6xz\) thus

\(2xy+4x^2-6xz =2x(y+2x-3z)\)