Partial Fractions
What are partial fractions?
- Partial fractions allow us to simplify rational expressions into the sum of two or more fractions with constant numerators and linear denominators
- This allows for integration of rational functions
- The method of partial fractions is essentially the reverse of adding or subtracting fractions
- When adding fractions, a common denominator is required
- In partial fractions the common denominator is split into parts (factors)
- If we have a rational function with a quadratic on the denominator partial fractions can be used to rewrite it as the sum of two rational functions with linear denominators
- This works if the non-linear denominator can be factorised into two distinct factors
- For example:
- This works if the non-linear denominator can be factorised into two distinct factors
- If we have a rational function with a linear numerator and denominator partial fractions can be used to rewrite it as the sum of a constant and a fraction with a linear denominator
- The linear denominator does not need to be factorised
- For example:
How do I find partial fractions if the denominator is a quadratic?
- STEP 1
Factorise the denominator into the product of two linear factors - Check the numerator and cancel out any common factors
- e.g.
- STEP 2
Split the fraction into a sum of two fractions with single linear denominators each having unknown constant numerators - Use A and B to represent the unknown numerators
- e.g.
- STEP 3
Multiply through by the denominator to eliminate fractions - Eliminate fractions by cancelling all common expressions
- e.g.
- STEP 4
Substitute values into the identity and solve for the unknown constants - Use the root of each linear factor as a value of to find the unknowns
- e.g. Let x = 2: etc
- An alternative method is comparing coefficients
- e.g.
- STEP 5
Write the original as partial fractions - Substitute the values you found for A and B into your expression from STEP 2
- e.g.
How do I find partial fractions if the numerator and denominator are both linear?
- If the denominator is not a quadratic expression you will be given the form in which the partial fractions should be expressed
- For example express in the form
- STEP 1
Multiply through by the denominator to eliminate fractions- e.g.
- STEP 2
Expand the expression on the right-hand side and compare coefficients- Compare the coefficients of x and solve for the first unknown
- e.g. 12x = 3Ax
- therefore A = 4
- Compare the constant coefficients and solve for the second unknown
-
- e.g. - 2 = - A + B = - 4 + B
- therefore B = 2
-
- Compare the coefficients of x and solve for the first unknown
- STEP 3
Write the original as partial fractions
How do I find partial fractions if the denominator has a squared linear term?
- A squared linear factor in the denominator actually represents two factors rather than one
- This must be taken into account when the rational function is split into partial fractions
- For the squared linear denominator (ax + b)2 there will be two factors: (ax + b) and (ax + b)2
- So the rational expression becomes
- In IB you will be given the form into which you should split the partial fractions
- Put the rational expression equal to the given form and then continue with the steps above
- There is more than one way of finding the missing values when working with partial fractions
- Substituting values is usually quickest, however you should look at the number of times a bracket is repeated to help you decide which method to use
Exam Tip
- An exam question will often have partial fractions as part (a) and then integration or using the binomial theorem as part (b)
- Make sure you use your partial fractions found in part (a) to answer the next part of the question
Worked Example
a)
Express in partial fractions.
b)
Express in the form .