Binomial expansion, explained with examples

By The STEM-Tutor team · Updated October 2026

The binomial expansion is a quick way to expand brackets like (1 + x)⁸ without multiplying them out by hand. It is in both years of A level Maths.

Pascal's triangle

Expanding (a + b)ⁿ for small n shows a pattern in the coefficients:

• n = 1: 1, 1
• n = 2: 1, 2, 1
• n = 3: 1, 3, 3, 1
• n = 4: 1, 4, 6, 4, 1
• n = 5: 1, 5, 10, 10, 5, 1

Each number is the sum of the two above it. This is Pascal's triangle. It works well for small powers, but for (1 + x)¹² you need a faster method.

The binomial formula for positive whole powers

(a + b)ⁿ = aⁿ + ⁿC₁aⁿ⁻¹b + ⁿC₂aⁿ⁻²b² + … + ⁿC_ra^n−rb^r + … + bⁿ

The coefficients ⁿC_r = n!/(r!(n − r)!) are the numbers in Pascal's triangle. Notice the pattern: the power of a goes down by 1 each term while the power of b goes up by 1, and the two powers always add up to n. There are n + 1 terms.

Why the coefficients are nCr

When you multiply out (a + b)ⁿ, you choose either a or b from each of the n brackets and multiply your choices together. A term containing b^r comes from choosing b in exactly r of the n brackets. The number of ways to choose those r brackets is ⁿC_r, which is why it appears as the coefficient. Your calculator has an nCr button: for example, 7 nCr 3 gives 35. This also explains why Pascal's triangle is symmetric: choosing r brackets for b is the same as choosing n − r brackets for a.

Worked examples

Example 1: a full expansion

Expand (1 + 2x)⁴.

• Coefficients from row 4: 1, 4, 6, 4, 1.
• 1 + 4(2x) + 6(2x)² + 4(2x)³ + (2x)⁴.
• = 1 + 8x + 24x² + 32x³ + 16x⁴. Put brackets around 2x so the whole term is raised to the power.

Example 2: the first few terms

Find the first three terms, in ascending powers of x, of (2 − x)⁶.

• 2⁶ + ⁶C₁(2⁵)(−x) + ⁶C₂(2⁴)(−x)².
• = 64 + 6(32)(−x) + 15(16)x² = 64 − 192x + 240x².

Example 3: a specific coefficient

Find the coefficient of x³ in (3 + 2x)⁷.

• The x³ term has r = 3: ⁷C₃(3⁴)(2x)³.
• = 35 × 81 × 8x³ = 22,680x³. The coefficient is 22,680.

Example 4: finding an unknown

The coefficient of x² in (1 + kx)⁸ is 112. Find the possible values of k.

• The x² term is ⁸C₂(kx)² = 28k²x².
• 28k² = 112, so k² = 4 and k = ±2.

Example 5: an approximation

Use the first three terms of (1 + x)¹⁰ to estimate 1.01¹⁰.

• (1 + x)¹⁰ ≈ 1 + 10x + 45x². Put x = 0.01.
• ≈ 1 + 0.1 + 0.0045 = 1.1045. The true value is 1.10462…, so the estimate is very close because x is small.

Negative and fractional powers (Year 2)

When n is negative or a fraction, the expansion never stops: it becomes an infinite series. Use the general form

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

This series is only valid for |x| < 1. Outside that range, the terms grow instead of shrinking and the series does not converge. Exam questions almost always ask you to state the range of validity.

Example 6: a negative power

Expand (1 + x)⁻¹ up to x³.

• n = −1: 1 + (−1)x + (−1)(−2)x²/2 + (−1)(−2)(−3)x³/6.
• = 1 − x + x² − x³, valid for |x| < 1. This matches the geometric series 1/(1 + x).

Example 7: a fractional power

Expand √(1 + 4x) up to x², and state the range of validity.

• (1 + 4x)^1/2: n = 1/2 and the ‘x’ is 4x.
• 1 + (1/2)(4x) + (1/2)(−1/2)(4x)²/2 = 1 + 2x − 2x².
• Valid for |4x| < 1, so |x| < 1/4.

Example 8: when the first term is not 1

Expand (4 + x)^−1/2 up to x².

• Take out the 4: 4^−1/2(1 + x/4)^−1/2 = ½(1 + x/4)^−1/2.
• (1 + x/4)^−1/2 = 1 − x/8 + (3/8)(x/4)² = 1 − x/8 + 3x²/128.
• Multiply by ½: ½ − x/16 + 3x²/256, valid for |x| < 4.

Combining with partial fractions

A common Year 2 question asks you to expand a fraction such as (5x + 3)/((1 + x)(1 + 3x)). First split it into partial fractions, then expand each part with the binomial series and add. The overall range of validity is the narrower of the two ranges.

Common mistakes

• Forgetting brackets, so 2x⁴ is written instead of (2x)⁴ = 16x⁴.
• Losing negative signs, especially in (a − b)ⁿ.
• Using the series for (1 + x)ⁿ without first making the first term 1.
• Forgetting to state the range of validity for negative or fractional powers.

Practice questions

Try each one before opening the answer.

Frequently asked questions

Is the binomial expansion formula in the formula booklet?

Yes. Most boards give both the positive integer form and the general (1 + x)ⁿ series. You still need to practise using them accurately.

What is the difference between Year 1 and Year 2 binomial expansion?

Year 1 covers positive whole-number powers, which give finite expansions. Year 2 adds negative and fractional powers, which give infinite series with a range of validity.

Key takeaways

• Coefficients come from Pascal's triangle or ⁿC_r.
• Powers of a fall and powers of b rise; they add to n.
• Bracket the whole second term before raising it to a power.
• Negative and fractional powers give infinite series.
• State the range of validity: |x| < 1 for (1 + x)ⁿ.

Related guides

• Partial fractions
• A level Maths formula booklet
• A level Maths topics