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This topic is part of the TCS FREE high school mathematics 'How-to Library', and will help you to simplify more difficult algebraic 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.

Theory:

Example 1:

If one or more terms of an expression contain factors that are sums or differences in brackets, then the brackets may need to be removed by multiplication of factors before the expression can be simplified.

This example illustrates this situation.

24x - 4(12x - 10x + 2) + x

 First the brackets must be expanded, like this:

24x - 48x + 40x - 8 + x

Grouping like terms and simplifying we now have:

17x - 8

Example 2:

Rational expressions are simplified by removing (also called ‘cancelling’) common factors that appear in both the numerator and denominator of the expression. This means that both the numerator and denominator of a rational expression must be expressed as a product of factors (ie ‘factorised’) before the expression can be simplified.

For instance, this expression:

has the difference: ‘m² - m’ in the numerator, and the difference: ‘5m – 5’ in the denominator.

Before this expression can be simplified, both the numerator and denominator must be factorised, giving the following equivalent expression:

Now the expression can be simplified by cancelling the factor (m - 1) in both the numerator and denominator.

The simplified form of the original expression is then:

    

Example 3:

If an expression contains more than one term, and one or more of the terms are rational expressions similar to that in example 2 above, then you will need to:

·Â Â Â Â Â Â Â Â  factorise the expression as much as possible without combining the terms into a single term,

·Â Â Â Â Â Â Â Â  put all of the terms over a common denominator,

·Â Â Â Â Â Â Â Â  expand, simplify and factorise the numerator as necessary,

·Â Â Â Â Â Â Â Â  simplify the whole expression to cancel common factors.

 

For instance, the expression:

             

has two terms, both of which are rational expressions.

 

The denominators of both terms can be factorised, like this:

           

Writing this expression with a common denominator gives:

           

Now the numerator needs to be expanded, simplified, and factorised as necessary:

Expand the numerator:

           

 Simplify the numerator:

           

The numerator expression: a + 8 cannot be factorised further, so we are now ready to simplify.

Both the numerator and denominator expressions contain the factor (a+8), so this factor can be cancelled from both expressions, giving:

Method:

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 Algematics to simplify each of the example expressions above. Then you will be able to simplify similar expressions yourself.

Step 1  Enter the data

Click  and type your expression 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.

 

       Maths...

 

and then click

Step 2  Do preparatory steps

The steps used to prepare an expression for simplifying will depend on the kind of expression. The steps for each of the example expressions are described below. For other expressions you may have to experiment to find the expanding, factorising and simplifying steps that are most helpful.

Example 1:

24x - 4(12x - 10x + 2) + x

·Â Â Â Â Â Â Â Â  Click  (expand) to expand the second term, like this:

24x - 4´12x + 4´10x - 4´2 + x

Example 2:

           

We need to factorise the numerator and denominator separately, giving products of factors that can be cancelled.

·Â Â Â Â Â Â Â Â  To deal with the numerator and denominator as separate expressions, click the   (numerator)  (denominator) buttons so that both are DOWN.

·Â Â Â Â Â Â Â Â  Click  (factorise).

Now the expression has a product of factors in the numerator and denominator:

·Â Â Â Â Â Â Â Â  Click on the  and  buttons again so that both are up.

Example 3:

·Â Â Â Â Â Â Â Â  Click  (factorise) several times until the denominators of the two terms are completely factorised, like this:

If the two terms combine into a single term, you have gone too far. In that case, press Ctrl+Delete to delete the extra step.

·Â Â Â Â Â Â Â  Click  (common denominator) to put the two terms of this expression over a common denominator.

This gives the expression:

The numerator must be expanded, simplified and factorised as necessary.
(In this example, the factorise step is not required.)

·Â Â Â Â Â Â Â Â  Click  (numerator) so that it is DOWN, to operate only on the numerator.

·Â Â Â Â Â Â Â Â  Click  (expand):

·Â Â Â Â Â Â Â Â  Click  (simplify):

The numerator has reduced to the expression: ‘a + 8’ which is a common factor in both the numerator and denominator and so can be cancelled.

·Â Â Â Â Â Â Â Â  Turn off the numerator command when you are finished simplifying by clicking on the  button again.

Step 3  Use the Simplify command

·Â Â Â Â Â Â Â Â  Click  (simplify) repeatedly until the expression is completely simplified.

If the expression is not changed by the simplify command, then it is completely simplified. If you duplicate the last step, you can delete it by using Ctrl+Delete on the keyboard.