Binomial Expansion Exam Questions: Practice + Worked Solutions

If you’ve just been handed a binomial expansion exam question and you’re not sure where to start, the fastest fix is this: identify whether n is a positive whole number or not, then pick the matching formula the finite binomial theorem for positive integer powers, or the infinite series for rational and negative powers. Get that one decision right and most of the marks follow.

That single fork in the road is where most students lose marks  not because the algebra is hard, but because they reach for the wrong version of the formula. This guide walks through both cases with fully worked binomial expansion exam questions, the mistakes examiners see most often, and a bank of practice problems with solutions so you can check your own working before the real thing.

The General Binomial Expansion Formula

For a positive integer n, the binomial expansion of (a + b)ⁿ is built from the binomial coefficients  the same numbers that appear in Pascal’s triangle. Each term follows the pattern C(n,k)·aⁿ⁻ᵏ·bᵏ, and the full expansion has exactly n + 1 terms, starting and ending with the coefficient 1.

The binomial expansion exam questions coefficient C(n,k), often written as ⁿCₖ or “n choose k,” is calculated as n! / (k!(n−k)!). You don’t need to memorize factorials under exam pressure, though  Pascal’s triangle gives you the same coefficients instantly for smaller values of n, and most exam boards provide the ⁿCₖ formula on the formula sheet anyway.

Worked example: Expand (1 + 2x)⁴ in ascending powers of x.

Using row 4 of Pascal’s triangle (1, 4, 6, 4, 1):

(1 + 2x)⁴ = 1 + 4(2x) + 6(2x)² + 4(2x)³ + (2x)⁴ = 1 + 8x + 24x² + 32x³ + 16x⁴

That’s the entire method for positive-integer powers: coefficients from Pascal’s triangle (or the ⁿCₖ formula), then powers of each term rising and falling in step.

Binomial Expansion for Rational and Negative Powers

This is where most binomial expansion exam questions at A-Level actually get harder, and it’s the single most commonly tested variant on Edexcel, AQA, and OCR papers. When n is not a positive whole number  a fraction like ½ or ⅓, or a negative integer like −1 or −2 — the expansion no longer terminates. It becomes an infinite series, and it’s only valid within a specific range of x.

The formula for (1 + x)ⁿ when n is rational becomes:

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

This binomial expansion exam questions is only valid when |x| < 1. If your question uses (a + bx)ⁿ instead of (1 + bx)ⁿ, you must first factor out aⁿ before applying the series  forgetting this factoring step is the single most common error examiners flag in mark schemes for this topic. The validity condition also shifts to |bx/a| < 1, not just |x| < 1, once you’ve factored.

Worked example: Find the first four terms of the expansion of (1 − 3x)⁻², stating the range of values of x for which it is valid.

Here n = −2, so:

(1 − 3x)⁻² = 1 + (−2)(−3x) + (−2)(−3)/2!² + (−2)(−3)(−4)/3!³ + … = 1 + 6x + 27x² + 108x³ + …

Valid for |−3x| < 1, i.e. |x| < 1/3.

Finding a Specific Coefficient (the most common exam-question type)

A large share of binomial expansion exam questions don’t ask for the full expansion at all  they ask for one specific term, usually the coefficient of xᵏ for a given k. You don’t need to expand the whole thing; you only need the general term.

For (a + bx)ⁿ, the term containing xᵏ is: C(n,k)·aⁿ⁻ᵏ·(bx)ᵏ

Worked example: Find the coefficient of x³ in the expansion of (2 + 5x)⁷.

The x³ term is C(7,3)·2⁴·(5x)³ = 35 × 16 × 125x³ = 70,000x³

So the coefficient is 70,000. This shortcut  going straight to the relevant term instead of expanding everything  is worth practicing until it’s automatic, since it’s faster and less error-prone under time pressure.

Using Binomial Expansions for Approximation

Exam boards frequently follow an expansion question with a part (b) that asks you to substitute a value of x to approximate a number like (1.02)⁸ or √1.04 without a calculator. This tests whether you understand why the expansion works, not just how to generate it.

Worked example: Use the expansion of (1 + x/4)⁸ to estimate (1.025)⁸ to 4 decimal places.

First expand (1 + x/4)⁸ = 1 + 2x + 1.75x² + … (first three terms, using ⁿCₖ with n = 8). Since 1.025 = 1 + 0.1/4, substitute x = 0.1:

1 + 2(0.1) + 1.75(0.1)² ≈ 1 + 0.2 + 0.0175 = 1.2175

This kind of two-part question  expand, then substitute is a near-guaranteed feature of Edexcel C2/Pure Maths papers on this topic, so it’s worth practicing as a pair rather than treating expansion and approximation as separate skills.

Practice Questions with Full Solutions

1. Find the first three terms, in ascending powers of x, of the expansion of (2 − 5x)⁵. Solution: C(5,0)·2⁵ + C(5,1)·2⁴·(−5x) + C(5,2)·2³·(−5x)² = 32 − 400x + 2000x² − …

2. Find the coefficient of x⁴ in the expansion of (5x − 2)⁹. Solution: The x⁴ term is C(9,4)·(5x)⁴·(−2)⁵ = 126 × 625x⁴ × (−32) = −2,520,000x⁴

3. Expand (1 + x)⁻¹ in ascending powers of x up to the term in x³, and state the range of validity. Solution: 1 − x + x² − x³ + …, valid for |x| < 1

4. A level maths practice: find the value of k such that the coefficient of x³ in (2 + kx)⁷ is 1890. Solution: The x³ term is C(7,3)·2⁴·(kx)³ = 35 × 16 × k³x³ = 560k³x³. Setting 560k³ = 1890 gives k³ = 3.375, so k = 1.5.

Comparison: Exam Boards and What They Test

All three UK boards teach the finite expansion first (positive integer n) and the infinite series for rational/negative n a year later, so double-check which version your specific paper is testing before you revise  it’s the most common source of wasted revision time on this topic.

Conclusion

Binomial expansion exam questions can seem challenging at first, but they become much easier once you understand the key patterns and methods. Practicing different question types helps you become confident with coefficients, specific terms, powers, and approximations. It is also useful to show each step clearly, as small mistakes in signs, powers, or calculations can affect the final answer.

 Regular practice with binomial expansion exam questions can improve both speed and accuracy while helping you recognize what a question is really asking. Keep revisiting the areas you find difficult and learn from your mistakes. With steady practice and a clear understanding of the binomial theorem, you can approach your next exam with greater confidence.

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