Other factorising techniques
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How to factorise several kinds of algebra expressions.

This topic is part of the TCS FREE high school mathematics 'How-to Library', and will help you to factorise several kinds of algebra expressions.   
(See the index page for a list of all available topics in the library.) To make best use of this topic, you need to download the Algematics software. Click here for instructions.


Algematics uses six different methods to factorise an algebraic expression. They are:

1. common factors

8p² + 12pq - 4pr  factorises to:  4p(2p + 3q - r)

2. difference of perfect squares

a² - b²  factorises to:  (a + b)(a - b)

3. sum of perfect cubes

a³ + b³  factorises to:  (a + b)(a² - ab + b²)

4. difference of perfect cubes

a³ - b³  factorises to:  (a - b)(a² + ab + b²)

5. split middle term of quadratics

2x² + 3x - 5  becomes:  2x² - 2x + 5x - 5

(While this has not factorised the expressions, this is just the first of several steps needed to factorise the quadratic. In this case you will need to use the factorise command two more times. )

[ To find out how to split the middle term of a quadratic yourself, click here. ]

6. group and factorise four terms

2x² - 2x + 5x - 5  becomes:  2x(x - 1) + 5(x - 1)

This does two things. First, the four terms are swapped around and regrouped if necessary, then the pairs of terms are factorised in such a way that a common factor results. In this example, (x-1) is now a common factor, so that if the factorise command is used one more time the expression will be fully factorised.

The software examines the expression and chooses the method to use. All you have to do is keep on clicking the  (factorise) button until the expression no longer changes.

If you click  and nothing changes, then the software cannot factorise the expression. Press Ctrl+Delete to delete the duplicate step.

Download the free support file... We have created an Algematics document containing the completed examples from this topic. It also includes practice exercises to improve your skills.

File name:  'Other factorising techniques.alg'   File size: 6kb
Click here
to download the file.

If you choose 'Open this file from its current location', then Algematics should open the document immediately. If not, try the other option: 'Save this file to disk', then run Algematics and choose the 'Open' command from the 'File' menu. Locate the saved file and open it. If you do not yet have Algematics installed on your computer, click here for instructions.


IMPORTANT: This topic assumes that you know how to enter mathematical formulas into Algematics. Find out how by completing the three simple tutorials in the 'Getting Started' section of the Algematics program 'Help'.

We will use the expression: to demonstrate how to use the factorise command.

Step 1  Enter the data

Click  and type the expression to be factorised into the maths box in the
data entry dialog box.

If the ‘EMPTY’ message is not displayed between the blue buttons, click the  button until the message: ‘INSERT’ appears. If  required, use the ‘ * ’ symbol for multiply, and the ‘ / ’ symbol for divide.


   x[6] - 1




and then click

Step 2  Factorise

Click the  (factorise) button. This will generate a new step.

If this new step is the same as the last, then this means that Algematics cannot factorise the expression any further. In that case, delete the duplicate step by pressing Ctrl+Delete on the keyboard.

Keep clicking  until the expression cannot be factorised any further.

The example expression factorises in the following sequence:

Click  : (Difference of perfect squares)

(x³ + 1)(x³ - 1)

Click  : (Sum of perfect cubes)

(x + 1)(x² - x´1 + 1)(x³ - 1)

Click  : (Difference of perfect cubes)

(x + 1)(x² - x´1 + 1)(x - 1)(x² + x´1 + 1)

This expression is now fully factorised, and can be tidied up by clicking  (simplify all), giving:

(x + 1)(x² - x + 1)(x - 1)(x² + x + 1)

Go back to step 1 to factorise another expression.


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