Partial Fractions
Neil Trivedi
Teacher
Partial Fractions
We are very comfortable with adding algebraic fractions, such as, showing that
However, what if we are already given and we want to split into smaller parts? If that is the case, we'll need a technique to work in reverse.
Example 1:
Write in the form using partial fractions, where and are constants to be found.
Note: The numerator must have a degree (highest power of a polynomial) that is lower than the denominator for this method to work.
Step 1: Write the RHS as a single fraction to match the same form as the LHS by
cross-multiplying.
Step 2: Equate the numerators and solve for and by substituting suitable values for that make one of the brackets equal to
Note: You can substitute any value of as the LHS is equivalent to the RHS meaning they are equal for all values of Therefore, it is easier to substitute values of that removes one of the constants.
Therefore, we have
No answer provided.
Example 2:
Write in partial fractions.
Single Step: As we get more comfortable rewriting the RHS into a single fraction, we can skip Step 1 shown in Example 1 and go straight to equating the numerators. We then solve for and by substituting suitable values for that make one of the brackets equal to
Therefore, we have
No answer provided.
Repeated Factors
If we want to represent something like in partial fractions, we can't write it merely as
because both terms have the same denominator. They would combine to to produce a fraction with a common denominator, which is not equivalent to the original fraction.
When we have repeated factors, such as in the denominator, we need to express the partial fractions as where we consider both the squared factor and its root equivalent.
We can illustrate this idea with normal numbers, such as We have which can also be written as
Example 3:
Write in partial fractions.
Step 1: Write the RHS as a single fraction. We want the denominator to be So, we do not cross-multiply in this case. Instead, we multiply the numerator and the denominator of by We don’t need to do anything to because its denominator is already We get:
Step 2: Equate the numerators and solve for and by substituting suitable values for that make one of the brackets equal to
However, notice that it is only possible to make the bracket equal to to find The next step is to substitute any value of (due to equivalence) and the value of to find
The value of you substitute is entirely up to you but sticking to easy substitutions, such as makes the most sense and is what we will be using.
Therefore, we have
No answer provided.
Improper Fractions
If the degree (highest power of a polynomial) of the numerator is equal to or larger than that of the denominator, we call it an improper algebraic fraction.
For example, take The degree of the numerator and denominator is so this is an improper algebraic fraction.
We deal with this in a similar way to normal fractions. For example, we can write as since goes into six times with a remainder of We use long division to find this, and the same method applies to improper algebraic fractions.
Example 4:
Express in partial fractions.
Step 1: As mentioned earlier, the degree of the numerator and denominator of this fraction is so this is an improper fraction. First, we use long division.
goes into three times so we put at the top and multiply and to give which we write underneath and subtract to give as our remainder.
So our improper algebraic fraction can now be written as:
Step 2: Now, we express in partial fractions. Write the RHS as a single fraction, then equate the numerators and solve for and by substituting suitable values for to make one of the brackets equal to
Therefore, we have
No answer provided.
Practice Questions