2 Proper fractions with linear factors
Firstly we describe how to calculate partial fractions for proper fractions where the denominator may be written as a product of linear factors. The steps are as follows:
Factorise the denominator.
Each factor will produce a partial fraction. A factor such as will produce a partial
fraction of the form where is an unknown constant. In general a linear factor
will produce a partial fraction . The unknown constants for each partial
fraction may be different and so we will call them , , and so on.
Evaluate the unknown constants by equating coefficients or using specific values of .
The sum of the partial fractions is identical to the original algebraic fraction for all values of .
Key Point 14
A linear factor in the denominator gives rise to a single partial fraction of the form
The steps involved in expressing a proper fraction as partial fractions are illustrated in the following Example.
Example 41
Express in terms of partial fractions.
Solution
Note that this fraction is proper. The denominator is factorised to give . Each of the linear factors produces a partial fraction. The factor produces a partial fraction of the form and the factor produces a partial fraction , where and are constants which we need to find. We write
By multiplying both sides by we obtain
…(*)
We may now let take any value we choose . By an appropriate choice we can simplify the right-hand side. Let because this choice eliminates . We find
so that the constant must equal 3. The constant can be found either by substituting some other value for or alternatively by ‘equating coefficients’.
Observe that, by rearranging the right-hand side, Equation (*) can be written as
Comparing the coefficients of on both sides we see that . We already know and so
from which . We can therefore write
We have succeeded in expressing the given fraction as the sum of its partial fractions. The result can always be checked by adding the fractions on the right.
Task!
Express in partial fractions.
First factorise the denominator:
Because there are two linear factors we write
Multiply both sides by to obtain the equation from which to find and :
Substitute an appropriate value for to obtain :
Substitute and get
Equating coefficients of to obtain the value of :
, since
Finally, write down the partial fractions:
Exercises
-
Find the partial fractions of
- ,
- ,
-
.
Check by adding the partial fractions together again.
-
Express each of the following as the sum of partial fractions:
- ,
- ,
- ,
1(a) , 1(b) 1(c) ,
2(a) , 2(b) , 2(c) .