1.1.3 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:
• 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
• 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