The binomial expansion · Year 13

In Year 13 you learn: binomial coefficients.

The binomial expansion is a quick way to expand (a + b)ⁿ without multiplying out bracket after bracket. The coefficients come from Pascal’s triangle or from ⁿCᵣ, and the method extends to approximations and to powers that are not whole numbers.

Key facts

• (a + b)ⁿ = aⁿ + ⁿC₁aⁿ⁻¹b + ⁿC₂aⁿ⁻²b² + … + bⁿ for whole-number n.
• ⁿCᵣ = (n!)/(r!(n − r)!): the numbers in row n of Pascal’s triangle.
• The general term is ⁿCᵣ aⁿ⁻ʳbʳ; it is the term containing bʳ.
• Keep signs and numbers inside the bracket: in (2 − 3x)⁵, b is −3x.

Worked examples

Other years

• The binomial expansion · Year 12