Binomial Expansion (CIE A Level Maths: Pure 1)

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Binomial Expansion

What is the binomial expansion?

Binomial Expansion Notes Diagram 1, A Level & AS Level Pure Maths Revision Notes

  • Look at the pattern
    • Start at nC0, then nC1, nC2, etc
    • Powers of a start at n and decrease by 1
    • Powers of b start at 0 and increase by 1

  • There are shortcuts but these hide the pattern
    • nC0 = nCn = 1
    • nC1 = nCn-= n
    • nCr = nCn-r
    • (b)0 = (a)0 = 1

  • Use the shortcuts once familiar with the pattern
  • ! means factorial

Binomial Expansion Notes Diagram 2, A Level & AS Level Pure Maths Revision Notes 

  • This is given in the formula booklet

What about Pascal’s triangle?

Binomial Expansion Notes Diagram 3, A Level & AS Level Pure Maths Revision Notes

  • Pascal’s triangle is an alternative to nCr
  • It is useful for lower values of n
  • For larger n it is slow and prone to arithmetic errors

How do I expand brackets with binomial expansion?

Binomial Expansion Notes Diagram 4, A Level & AS Level Pure Maths Revision Notes

  • Use a line for each term to make things easier to read and follow
  • Use brackets, particularly helpful when negatives involved
  • Use a calculator for nCr

Binomial Expansion Notes Diagram 5, A Level & AS Level Pure Maths Revision Notes

Worked example

Binomial Expansion Example Diagram, A Level & AS Level Pure Maths Revision Notes

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Applications of Binomial Expansion

What are binomial expansions used for?

  • Expanding brackets, it is usual to be asked for either …
    • … the first few terms only

     Binomial Expansion Notes Diagram 6, A Level & AS Level Pure Maths Revision Notes

  • or (the coefficient of) a particular term

 

Binomial Expansion Notes Diagram 7, A Level & AS Level Pure Maths Revision Notes

  • Solving problems in unknowns

Binomial Expansion Notes Diagram 8, 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.