Page accessed [ ] times since 1 June 2006
Page written by Judy Edwards, Thomasin Smith and Kee Teo with assistance from Rebecca Keen.
You are probably familiar with the process of "multiplying out", that is to say, using the distributive law to express a product as a sum of terms. For example: 3(x + 2) = 3(x) + 3(2) = 3x + 6.
Sometimes, especially when solving equations, it is useful to reverse the process of "multiplying out", that is to say, to express the sum of a number of terms as a product. This process is called factorisation. There are a number of ways of doing this, but by far the most important in practice is what is called "taking out a common factor".
Essentially, this is just applying the distributive law in reverse. For example, multiplying out gives 3(x + 2) = 3(x) + 3(2) = 3x + 6. Looking at it the other way round, suppose we start with the expression 3x + 6. Notice that since 6 = 3(2), we can write the expression as 3(x) + 3(2), where 3 is a factor of both terms (we call it a common factor of the terms). Now we can apply the distributive law in reverse:
3x + 6 = 3(x) +3(2) = 3(x + 2)
and so we have factorised the expression by taking the common factor 3 and putting it outside the brackets. A factor does not have to be an explicit number, it may be a variable (letter x in this case).
Have a go at these now.
Now you can use your factor-spotting skills to factorise expressions. Here is one more example to remind you how it works:
Sometimes a common factor may itself be a sum of terms.
Here are a few like this for you to practice on:
Common factors may also involve functions such as trigonometric or exponential functions.