Binomial Expansion (Edexcel IAL Maths)

Spec point — 4.5

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
  • ⁿ**Cr = **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 is 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 the 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

Worked Example

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

• or (the coefficient of) a particular term

• Solving problems in unknowns