Simplifying Algebraic Fractions (common factors)

This chapter explores simplifying algebraic fractions by cancelling common factors. The same concept applies as when working with simple numerical fractions you cancel down by finding factors that are common to both the numerator and the denominator. In the following examples we shall see that the same applies.

Example 1

This example consists of a simple numerical fraction. Suppose we had to work out;
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First we can expand it or find a factor that is common to both numerator and denominator and cancel it out as shown below.
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In this following example we shall look at a simple algebraic fraction. Suppose we had to work out;
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We will need to cancel the expression down but to do that we will need to write both the numerator and the denominator as products. We need to find the common product for both the numerator and denominator. This common factor is (x + 3).
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…that leaves a half as the answer.
You can’t however do the following. It is wrong. You can’t cancel over addition;
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The above is wrong and don’t attempt it as the outcome will be wrong in your calculation.
You may come across a problem where you have fractions in both the numerator and denominator of a fraction. It is common sense to create an equivalent fraction by multiplying out by the same number as shown below. For example suppose we had to work out the following;
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Fir we multiply the numerator and denominator by 6 as shown below.
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…this gives the following;
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…to simplify the expression we will need to factorise the numerator and denominator as shown below;
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Now we can cancel out the common factor;
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Example 2

In these following examples we shall look at more complex algebraic fractions; Suppose we had to work out;
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Again as we have been doing in the above examples we will need to find the common factor for the numerator and denominator;
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…we cancel out the factors;
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…that leaves the following as the answer;
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In the following example there is a fraction in the numerator which we will have to get rid of;
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To remove the fraction we multiply the numerator and denominator by x as shown below;
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…this gives;
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…we will need to factorise the expressions cancel out the common factors as shown below;
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The final answer becomes;
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