Partial Fractions (DP IB Analysis & Approaches (AA)): Revision Note

Partial Fractions

What are partial fractions?

• Partial fractions are the reverse of adding or subtracting algebraic fractions

  • e.g. in partial fractions is

    • This reverses

How do I find partial fractions when the denominator is quadratic?

• STEP 1
  Factorise the denominator into two linear factors (and simplify if necessary)

  • e.g. 

• STEP 2
  Split the fraction into the sum of two smaller fractions with linear denominators and unknown constant numerators

  • Use and to represent the unknown numerators

    • e.g.  

• STEP 3
  Multiply both sides by the denominator to eliminate fractions

  • e.g.

  • which gives 

• STEP 4
  Find the unknown constants by substituting different numerical values into both sides to form and solve simultaneous equations in and

  • The easiest equations come from substituting in the roots of each linear factor

    • e.g. let : 

    • so giving

    • Let :

    • so giving

  • An alternative method is comparing coefficients

    • e.g. write it as  

    • The coefficient of on both sides must match, so

    • The constant term on both sides must match, so

    • Then solve these simultaneously

• STEP 5
  Write the original fraction in partial fractions

  • Substitute and back into the expression

    • e.g.  

• In general, if the denominator factorises then

  • 

How do I find partial fractions when the denominator is the square of a linear term?

• A squared linear factor in the denominator must be split into two different partial fractions of the form

  • This can be seen in reverse by trying to add, for example,

    • The lowest common denominator is

    • 

• In general,

  • Then use the same method as above

How do I find partial fractions when both the numerator and denominator are linear?

• If the numerator and denominator are both linear, the fraction can be split into a constant and a simpler fraction

  • 

    • Then use the same method as above

• e.g. write as to get

  • Substitute in to get

  • Substitute in, say, to get so

    •