The binomial expansion

The binomial expansion multiplies out (a + b)ⁿ without doing all the brackets. The coefficients come from Pascal’s triangle or nCr.

Key facts

  • (a + b)ⁿ = Σ nCr aⁿ⁻ʳ bʳ
  • Pascal’s triangle: each number is the sum of the two above.
  • The term in xʳ of (1 + kx)ⁿ is nCr kʳ xʳ.
  • For negative or fractional n (A-level): (1 + x)ⁿ = 1 + nx + n(n − 1)/2! x² + … for |x| < 1.

Worked examples

Example 1

Expand (1 + 2x)⁴.

  1. Row 4: 1, 4, 6, 4, 1.
  2. Multiply each by the matching power of 2x. 1 + 4(2x) + 6(2x)² + 4(2x)³ + (2x)⁴
  3. = 1 + 8x + 24x² + 32x³ + 16x⁴

Answer: 1 + 8x + 24x² + 32x³ + 16x⁴

Example 2

Find the coefficient of x³ in (2 + x)⁷.

  1. The x³ term is 7C3 × 2⁴ × x³.
  2. 35 × 16 = 560

Answer: 560