IB DP Maths: AA HL复习笔记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
• 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

How do I find partial fractions if the denominator is a quadratic?

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
• 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
• 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