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

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General Binomial Expansion

What is the general binomial expansion?

  • The binomial expansion applies for positive integers, n
  • The general binomial expansion applies to other types of powers too

GBE Notes Diagram 1, A Level & AS Level Pure Maths Revision Notes

  • The general binomial expansion applies for all real numbers,
  • Usually fractional and/or negative values of n are used
  • It is derived from (a + b)n, with a = 1 and b = x
  • a = 1 is the main reason the expansion can be reduced so much

GBE Notes Diagram 2, A Level & AS Level Pure Maths Revision Notes

  • Unless n , the expansion is infinitely long
  • It is only valid for |x| < 1
    • This is another way of writing -1 < x < 1
    • This is often called the validity statement
    • The restriction |x| < 1 means the series will converge
    • Higher powers of x can be ignored (as r , xr 0)
    • Only the first few terms of an expansion are needed

How do I expand brackets with the general binomial expansion?

STEP 1       Write the expression in the form (1 + x)n
STEP 2       Expand and simplify

Use a line for each term to make things easier to read and follow

Use brackets - fractions and negatives get ugly!

STEP 3       If required, check and state the validity statement

GBE Notes Diagram 3, A Level & AS Level Pure Maths Revision Notes

How do I use the general binomial expansion when it is (1 + bx)?

STEP 1       Write the expression in the form (1 + bx)n
STEP 2       Replace “x” by “bx” in the expansion

Check carefully to see if b is negative

STEP 3       Expand and simplify

Use a line for each term to make things easier to read and follow

Use brackets

STEP 4       If required, check and state the validity statement
      The validity statement changes   

      Replace “x” with “bx” so now |bx| < 1

 GBE Notes Diagram 4, A Level & AS Level Pure Maths Revision Notes

GBE Notes Diagram 5, A Level & AS Level Pure Maths Revision Notes

 

Worked example

GBE Example Diagram 1, A Level & AS Level Pure Maths Revision NotesGBE 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.