The Binomial Expansion

The binomial expansion

The binomial expansion expands (a + b)ⁿ without multiplying out all the brackets. For a positive integer n:

(a + b)ⁿ = Σ nCr · a^(n−r) · bʳ   (r = 0 to n)

where nCr = n! ÷ [r!(n−r)!] are the binomial coefficients (also read from Pascal's triangle).

Pascal's triangle

Each row gives the coefficients; each number is the sum of the two above:

1
1 1
1 2 1
1 3 3 1
1 4 6 4 1

Row n gives the coefficients of (a + b)ⁿ.

Example expansion

(1 + x)⁴ = 1 + 4x + 6x² + 4x³ + x⁴ (coefficients from row 4).

For (2 + 3x)³:

= 2³ + 3(2²)(3x) + 3(2)(3x)² + (3x)³
= 8 + 36x + 54x² + 27x³

Finding a specific term

The term in bʳ is nCr · a^(n−r) · bʳ. You don't need the whole expansion — pick the r that gives the power you want.

Expansion for any n (|x| < 1)

For fractional or negative n, the expansion is an infinite series, valid only when |x| < 1:

(1 + x)ⁿ = 1 + nx + [n(n−1)/2!]x² + [n(n−1)(n−2)/3!]x³ + …

For (a + bx)ⁿ, factor out a first: (a + bx)ⁿ = aⁿ(1 + (b/a)x)ⁿ, valid for |(b/a)x| < 1. Used for approximations and series.

Worked example

Find the coefficient of x² in the expansion of (1 + 2x)⁵.

• Term = 5C2 · 1³ · (2x)² = 10 × 4x² = 40x². Coefficient = 40. ✓

Common mistakes

• Forgetting to raise the whole term to the power (e.g. (3x)² = 9x², not 3x²).
• Using the wrong row of Pascal's triangle (row n for power n).
• Forgetting the validity condition |x| < 1 for fractional/negative n.

Exam tips

• For a single term, use nCr a^(n−r) bʳ — no need to expand everything.
• Factor out the constant for (a + bx)ⁿ before using the general expansion.
• State the range of validity for fractional/negative powers.

Key facts to remember

• (a + b)ⁿ = Σ nCr a^(n−r) bʳ; coefficients from Pascal's triangle (positive integer n).
• General term in bʳ = nCr a^(n−r) bʳ.
• For fractional/negative n: infinite series valid only when |x| < 1 (factor out a first).