AQA A Level Maths: Pure复习笔记4.1.1 Binomial Expansion

Binomial Expansion

 

What is the binomial expansion?

4.1.1-Binomial-Expansion-Notes-Diagram-1

 

  • 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-1 = n
    • nCr = nCn-r
    • (b)0 = (a)0 = 1

     

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

4.1.1-Binomial-Expansion-Notes-Diagram-2

  • This is given in the formula booklet

What about Pascal’s triangle?

4.1.1-Binomial-Expansion-Notes-Diagram-3

  • 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?

4.1.1-Binomial-Expansion-Notes-Diagram-4

  • 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

4.1.1-Binomial-Expansion-Notes-Diagram-5

What are binomial expansions used for?

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

    4.1.1-Binomial-Expansion-Notes-Diagram-6

  • or (the coefficient of) a particular term

 

4.1.1-Binomial-Expansion-Notes-Diagram-7

  • Solving problems in unknowns

4.1.1-Binomial-Expansion-Notes-Diagram-8

 

Worked Example

4.1.1-Binomial-Expansion-Example-Diagram

转载自savemyexams

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