Linear Denominators (CIE A Level Maths: Pure 3)

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Linear Denominators

What are partial fractions?

Linear Denominators Notes Diagram 1, A Level & AS Level Pure Maths Revision Notes

 

  • This is the reverse process to adding (or subtracting) fractions
  • When adding fractions a common denominator is required
  • In partial fractions the common denominator is split into parts (factors)
  • Partial fractions are used in binomial expansions (see Multiple GBEs) and integration (see Integration by Parts)

What are linear denominators?

Linear Denominators Notes Diagram 2, A Level & AS Level Pure Maths Revision Notes

 

  • A linear factor is of the form (ax + b)
  • A non-linear denominator may be written as the product of linear factors
  • If the denominator can be factorised

 

Linear Denominators Notes Diagram 3, A Level & AS Level Pure Maths Revision Notes

How do I find partial fractions?

STEP 1        Factorise the polynomial in the denominator

(Sometimes the numerator can be factorised too)

STEP 2        Split the fraction into a sum with single linear denominators

STEP 3        Multiply by the denominator to get rid of fractions

STEP 4        Substitute values of x to find AB, etc

(An alternative method is comparing coefficients)

STEP 5        Write the original as partial fractionsLinear Denominators Notes Diagram 4, A Level & AS Level Pure Maths Revision Notes

 

Comparing coefficients

  • The quantity of each term must be equal on both sides
  • “The number of x2 on the LHS” = “The number of x2 on the RHS”
  • “The number of …” is called the coefficient of x2

Linear Denominators Notes Diagram 5, A Level & AS Level Pure Maths Revision Notes

Worked example

Linear Denominators Example Diagram, A Level & AS Level Pure Maths Revision Notes

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Paul

Author: Paul

Expertise: Maths

Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.