General Binomial Expansion - Multiple (CIE A Level Maths: Pure 3)

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General Binomial Expansion - Multiple

What is the general binomial expansion?

  • The general binomial expansion, as given in the formula booklet, is

 Multiple GBEs Notes Diagram 1, A Level & AS Level Pure Maths Revision Notes

  • If n then the expansion is finite (see Binomial Expansion)
  • Otherwise the expansion is infinitely long
    • It is only valid for |x| < 1 (-1 < x < 1)
    • Only the first few terms of an expansion are usually needed

What is meant by multiple general binomial expansions? 

  • More than one part of an expression can be a binomial expansion
  • These may sometimes be called compound expressions
  • The expansion will only be valid for the lowest |x| boundary from all the expansions used

How do I use general binomial expansions in complicated expressions?

STEP 1        Break the expression down into binomial expansions

STEP 2        Expand each binomial individually, up to a suitable number of terms

    • Be careful with negatives and fractions
    • Use brackets as appropriate

STEP 3        Collect the expansions together and simplify

    • This could be expanding brackets, collecting like terms, etc
    • Ignore any terms of degree higher than required

STEP 4        Check the validity of each binomial expansion

    • The overall validity is the intersection (∩)

Multiple GBEs Notes Diagram 4&5, A Level & AS Level Pure Maths Revision Notes

How do I work with partial fractions and the general binomial expansion?

  • Partial fractions allow rational expressions to be written in a form where the general binomial expansion can then be appliedMultiple GBEs EXTRA1, A Level & AS Level Pure Maths Revision Notes

  • Validity is an important part of the general binomial expansion

Multiple GBEs Notes Diagram 7, A Level & AS Level Pure Maths Revision Notes

Worked example

Multiple GBEs Example Diagram 1, A Level & AS Level Pure Maths Revision Notes

Multiple GBEs Example Diagram 2

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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.