Binomial expansion calculator with Pascal's triangle

The binomial theorem expands (a + b)ⁿ without multiplying the bracket out n times: (a + b)ⁿ = Σ nCr aⁿ⁻ʳ bʳ, with the coefficients nCr from Pascal's triangle. So (1 + 2x)⁴ = 1 + 8x + 24x² + 32x³ + 16x⁴. This binomial expansion calculator expands any bracket to a whole power exactly, finds a single term or coefficient, and expands negative and fractional powers as a series with the range it is valid for.

It opens on (2x − 3)⁵. Open the full binomial expansion calculator over this page to type your own.

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Binomial Expansion Calculator

Expand brackets to any power with Pascal's triangle

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How to use the binomial expansion calculator

1. Type a bracket to a power in Expression, such as (2x − 3)^5, (x + y)^6 or (1 + x)^(−2). Fractions work in the bracket and the power.
2. Read the expansion and the general term, with the working term by term. For whole powers up to 14, Pascal's triangle is drawn with the row you need picked out.
3. Under Find, type a term number or a power to pick out one term or the coefficient of that power.
4. For negative or fractional powers, choose how many terms of the series to show. The calculator gives the range of x it is valid for and graphs the series against the true curve.
5. Load an example from Examples, or press Random problem in the Practice section.

The binomial theorem

The powers of a go down from n to 0 while the powers of b go up from 0 to n, and every term's powers add to n. The coefficients are row n of Pascal's triangle, where each number is the sum of the two above it. Row 4 is 1, 4, 6, 4, 1. T_(r+1) is the general term: the (r + 1)th term, counting from 1.

Worked examples you can check in the calculator

Example 1: (1 + 2x)⁴

Row 4 of Pascal's triangle is 1, 4, 6, 4, 1, and here a = 1 and b = 2x:

Put the 2 inside the bracket before squaring or cubing: (2x)² is 4x², not 2x².

Example 2: a negative term, (2x − 3)⁵

With b = −3 the signs alternate. Row 5 is 1, 5, 10, 10, 5, 1, giving 32x⁵ − 240x⁴ + 720x³ − 1080x² + 810x − 243, the calculator's opening expansion. For example, the third term is 10 × (2x)³ × (−3)² = 10 × 8x³ × 9 = 720x³.

Example 3: one coefficient

Find the coefficient of x⁵ in (2x − 1)⁸. The general term is 8Cr (2x)⁸⁻ʳ (−1)ʳ. For x⁵ you need 8 − r = 5, so r = 3:

The coefficient is −1792. Type (2x − 1)^8 and put 5 in the power box under Find.

Example 4: a negative power

For a power that isn't a positive whole number, the expansion never ends, and it only works when the second term is small enough. The A-level formula book gives:

With n = −2: (1 + x)⁻² = 1 − 2x + 3x² − 4x³ + ..., valid for |x| < 1. The calculator's graph shows the series hugging the true curve near x = 0 and drifting away as |x| gets close to 1.

Example 5: a fractional power

For (4 + x)^(1/2), take out the 4 first: (4 + x)^(1/2) = 2(1 + x/4)^(1/2). Expanding with n = 1/2 and x/4 in place of x gives 2 + x/4 − x²/64 + x³/512 + ..., valid for |x| < 4. Putting x = 0.1 in gives 2.02485, matching √4.1 = 2.02485 to five decimal places.

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Common mistakes

• Leaving numbers outside the power. (2x)³ = 8x³, not 2x³.
• Dropping a minus sign. (−3)² = 9 but (−3)³ = −27, so the signs alternate.
• Off-by-one term numbers. The (r + 1)th term uses r, so the 4th term has r = 3.
• Not taking out a factor for a series. (4 + x)^(1/2) has to become 2(1 + x/4)^(1/2) first.
• Using a series outside its range. (1 + x)ⁿ for non-whole n only works for |x| < 1.

Read next

Maths · AlgebraBinomial expansion formula: (a + b)ⁿ with examplesThe binomial expansion formula for (a + b)ⁿ and (1 + x)ⁿ, Pascal's triangle, finding one coefficient, fractional and negative powers, with examples.