Example 1: Finding a coefficient
Find the coefficient of x² in the expansion of (1 + 2x)⁵.
- The
x²term isC(5, 2)(2x)² = 10 × 4x² = 40x²
Answer: 40
Binomial Expansion (DSE M1)
For a positive integer n, (a + b)ⁿ = Σ C(n, r) aⁿ⁻ʳ bʳ, r = 0 to n. M1 uses it mainly to pick out coefficients, often after multiplying by the series eˣ = 1 + x + x²/2! + ….
Binomial Expansion Formulas
Expansion
(a + b)ⁿ = C(n, 0)aⁿ + C(n, 1)aⁿ⁻¹b + … + C(n, n)bⁿSpecial case
(1 + x)ⁿ = 1 + nx + [n(n − 1)/2]x² + …Exponential series
ekx = 1 + kx + (kx)²/2! + (kx)³/3! + …Sigma notation
Σ(axr + byr) = aΣxr + bΣyrWorked Examples — Binomial Expansion
Example 2: A binomial expansion times an exponential series
Expand e⁻²ˣ(1 + x)⁵ in ascending powers of x as far as the term in x².
e⁻²ˣ = 1 − 2x + 2x² − …(1 + x)⁵ = 1 + 5x + 10x² + …- Constant:
1; x:5 − 2 = 3;x²:10 − 10 + 2 = 2
Answer: 1 + 3x + 2x² + …
💡 Exam Traps & Secrets
⚠️ Collect every product that gives x²When two series are multiplied, the
x² term comes from three products: constant × x², x × x, and x² × constant. Missing one is the usual error.⚠️ Square the whole of (2x)
(2x)² = 4x², not 2x². Keep the coefficient inside the bracket when raising to a power.✏️ Self-Test: Quick Interactive Practice
How to study: Click any question below to instantly load it on the coordinate grid and check your steps.
(Expansion)Expand (2 − x)⁴ in ascending powers of x.
(With an exponential series)Find the coefficient of x² in the expansion of e^(3x)(1 − x)⁴.
(Coefficient)Find the coefficient of x³ in the expansion of (1 + x/2)⁸.
Frequently Asked Questions
Expansion of trinomials, finding the greatest coefficient or greatest term, properties of binomial coefficients, and numerical approximation such as estimating 1.01 to the power 10.
It is the foundation for the binomial distribution, whose probabilities are the terms of the expansion of (p + q)ⁿ.
As many as the highest power asked for. To expand as far as x², keep the terms up to (kx)²/2!.