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