A-Level · A-Level Maths
A-Level Maths binomial expansion
What is binomial expansion?
Binomial expansion expands (a + b)^n. For positive integer n, the expansion is finite and uses (a + b)^n = Σ C(n,k) a^(n-k) b^k. For rational/negative n, the expansion is infinite and only valid when |b/a| < 1. Both forms are examinable across AQA, Edexcel and OCR. Sloppy work on the validity condition is a high-frequency lost mark.
Worked example
Expand (1 − 2x)^(1/2) up to and including x² and state the range of validity.
- Use the binomial series: (1 + y)^n = 1 + ny + n(n−1)/2! y² + ... with n = 1/2 and y = −2x.
- Term 1: 1.
- Term 2: ny = (1/2)(−2x) = −x.
- Term 3: n(n−1)/2! y² = (1/2)(−1/2)/2 · (−2x)² = (−1/8) · 4x² = −x²/2.
- So (1 − 2x)^(1/2) ≈ 1 − x − x²/2.
- Validity: |−2x| < 1 → |x| < 1/2.
Positive integer n: the finite expansion
When n is a positive integer, (a + b)^n has exactly n + 1 terms. The general term is C(n,k) · a^(n-k) · b^k. C(n,k) is the binomial coefficient ('n choose k'), often written as nCr on calculators.
Rational or negative n: the infinite expansion
When n is not a positive integer, (1 + y)^n becomes the infinite series 1 + ny + n(n-1)/2! y² + n(n-1)(n-2)/3! y³ + ... — and only converges when |y| < 1. Always state the range of validity in A-Level Maths.
Approximations
Binomial expansions are routinely used to find numerical approximations: e.g., (1.02)¹⁰ ≈ 1 + 10(0.02) + ... ≈ 1.219. Always state how many terms you used.
Common mistakes
- Forgetting to state the range of validity when n is rational or negative — a guaranteed 1-mark loss.
- Confusing the binomial theorem (positive integer) with the binomial series (any n).
- Errors in factorials: 3! = 6, not 9.
- Sign errors when the term inside the brackets is negative (−2x instead of +2x).
Frequently asked
- When does the binomial series converge?
- For (1 + y)^n with n not a positive integer, the series converges when |y| < 1. State this validity range explicitly in every answer.
- Do I need to memorise the formula?
- The general term and the binomial series are typically on the formula booklet at A-Level. You still need to know how to apply them — looking up the formula doesn't help if you can't substitute correctly.
- How is binomial expansion used in statistics?
- The binomial distribution probability mass function P(X = k) = C(n,k) · p^k · (1-p)^(n-k) uses the same binomial coefficient. The expansion of (p + q)^n is the sum of all probabilities and equals 1.
A-Level Maths glossary terms
- Binomial expansionBinomial expansion expands (a + b)^n using (a + b)^n = Σ (n choose k) a^(n−k) b^k for positive integer n. For rational/negative n the expansion is an infinite series, valid only for |b/a| < 1. A-Level Maths examines both forms; Further Maths extends to approximations. Sloppy work with the validity condition is a high-frequency lost mark.
- Chain ruleThe chain rule differentiates composite functions: if y = f(g(x)) then dy/dx = f'(g(x)) · g'(x). It appears across A-Level Maths and Further Maths whenever an outer function wraps an inner function — common examples are sin(2x), e^(3x²) and (1+x²)⁵. Without the chain rule, every composite differentiation drops marks; with it, the standard trick is to spot the inner function first.
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