Adding and Subtracting Fractions
Addition and subtraction of algebraic fractions works
basically the same way as does addition and subtraction of simple
numerical fractions:
(i.) rewrite all of the fractions as mathematically equivalent
fractions, but having the same denominator (ideally, the simplest
such common denominator).
(ii.) combine the numerators of these fractions as indicated
(adding and subtracting), retaining the common denominator.
When we work with algebraic fractions, there is then often a
third very important step:
(iii.) simplify the resulting fraction as much as possible. It
is always necessary to make sure that once the
addition/subtraction step is completed, you make a reasonable
effort to simplify the result. Usually this simplification step
will be made very much easier if you take care in step (i.) to
make sure that you have used the simplest possible common
denominator. We will illustrate the process for determining that
simplest possible denominator when the fractions to be
added/subtracted are algebraic fractions. You will see that the
strategy is essentially the same as the strategy used to
determine the least common denominator when adding or subtracting
numerical fractions, except that now symbolic factors may occur
as well as numerical factors.
Example 1:
Simplify: 
solution:
Here, the word “simplify” means “carry out the
indicated arithmetic and state the final result as a single
fraction in simplest form”. So, step (i.) is to determine
the simplest common denominator. To do this we factor each
denominator. Since all three denominators are just numbers, this
means factoring them into a product of prime factors:
2 = 2 1
4 = 2 2
and
14 = 2 · 7
Thus, the simplest common denominator must be 2 2
· 7 = 28, because it must contain a 2 2 (since 4 = 2
2 is the highest power of 2 occurring in the three
denominators), and it must contain 7 (since 7 is a factor of one
of the denominators.) Between 2 2 and 7, all three
denominators are accounted for.
So, now,
Since 31 is a prime number, we cannot find any common factors
between the numerator and denominator. So, the correct final
answer here is 
.
Example 2:
Simplify 
solution:
Factoring the denominators gives
10x = 2 · 5 · x
3x = 3 · x
and
4x = 2 2 · x
So, the simplest common denominator is 2 2 · 3 ·
5 · x = 60x. Thus
Since 17 is a prime number which does not divide evenly into
60, there are no common factors available for cancellation
between the numerator and denominator of this last fraction, so
this is the final answer in simplest form for this problem.
Example 3:
Simplify 
solution:
The denominators are already factored, since each is just a
simple power of x. The highest power of x occurring in a
denominator here is x 5, so this is the simplest
common denominator.
We get
We do not have any techniques for factoring a cubic polynomial
(which is what is present in the numerator of this last
fraction). However, here the only factors that could be cancelled
between the numerator and denominator to simplify this last
fraction would be a power of x, and it is clear that there is no
common power of x factor in the numerator and denominator.
Therefore, this last fraction cannot be simplified further and so
must be our final answer.
Example 4:
Simplify 
solution:
As written, the two denominators are factored. We see that the
simplest common denominator is 2 · 3 · t = 6t. So
The numerator of this last fraction cannot be factored
further, and so this last fraction cannot be simplified further.
It must be the final answer here.
Note that when the two fractions were combined over a common
denominator, the numerator was the difference of two terms, each
containing bracketed expressions. In this case, it was worthwhile
to try removing the brackets in the numerator to see if it could
be simplified by collecting like terms.
