CIE A Level Maths: Pure 1复习笔记4.1.1 Binomial Expansion

What is the binomial expansion?

• 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 = ⁿC_n = 1
  • ⁿC₁ = ⁿC_n-1 = n
  • ⁿCr = ⁿC_n-r
  • (b)⁰ = (a)⁰ = 1
• Use the shortcuts once familiar with the pattern
• ! means factorial

• This is given in the formula booklet

What about Pascal’s triangle?

• Pascal’s triangle is an alternative to ⁿC_r
• 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?

• 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 ⁿC_r

What are binomial expansions used for?

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

   

• or (the coefficient of) a particular term

• Solving problems in unknowns