A LevelMathsCambridge (CIE)Pure 3Revision NotesAlgebra & FunctionsPartial FractionsPartial Fractions with Quadratic Denominators

Partial Fractions with Quadratic Denominators (Cambridge (CIE) A Level Maths) : Revision Note

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Quadratic Denominators

What is meant by partial fractions with quadratic denominators?

  • For linear denominators the denominator of the original fraction can be factorised such that the denominator becomes a product of linear terms of the form left parenthesis a x space plus space b right parenthesis

  • With squared linear denominators, the same applies, except that some (usually just one) of the factors on the denominator may be squared, i.e. left parenthesis a x space plus space b right parenthesis squared

  • In both the above cases it can be shown that the numerators of each of the partial fractions will be a constant left parenthesis A comma space B comma space C comma space etc right parenthesis

  • For this course, quadratic denominators refer to fractions that have one linear factor and one quadratic factor (that cannot be factorised) on the denominator

    • the denominator of the quadratic partial fraction will be of the form left parenthesis a x squared space plus space b x space plus space c right parenthesis; very often b space equals space 0 leaving it as left parenthesis a x squared space plus space c right parenthesis

    • the numerator of the quadratic partial fraction could be of linear form, left parenthesis A x space plus space B right parenthesis

How do I find partial fractions involving quadratic denominators?

  •  STEP 1          Factorise the denominator as far as possible (if not already done so)

  • Sometimes the numerator can be factorised too

  • STEP 2          Split the fraction into a sum with

    • the linear denominator having an (unknown) constant numerator

    • the quadratic denominator having an (unknown) linear numerator

  • STEP 3          Multiply through by the denominator to eliminate fractions

  • STEP 4          Substitute values into the identity and solve for the unknown constants

    • Use the root of the linear factor as a value of x to find one of the unknowns

    • Use x space equals space 0 to find another one of the unknowns

    • Use any value of x (keep it small and simple) to find the final unknown

    • If the linear factor is then you'll need to use any two other values of  to form simultaneous equations

  • STEP 5          Write the original as partial fractions

  • In harder problems there may be more than one linear or quadratic factor

    • In such cases, values of x, whatever order they’re used in, will not always eliminate all but one of the unknowns

    • Simultaneous equations will need to be used

Worked Example

1-3-3-cie-fig1-we-solution-quad-den
1-3-3-cie-fig1-we-solution-quad-den_2

Examiner Tips and Tricks

  • You can check your final answer by substituting a value of x in to both the left and right-hand sides and seeing if they’re equal

    • Choose a small value of x  to keep things simple but not a value that would make a denominator zero

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