Enter the two terms and the power to expand (a + b)ⁿ. Terms can be numbers, fractions, decimals or variables with powers such as 3x², −2y or 1/x. You get the full expansion with exact fractional coefficients, a table of every term, the coefficients from Pascal's triangle, the middle term and any term number you ask for.
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e.g. x, 2x, 3x^2, x^2y, 1/2, -y
Use a minus sign for (a − b), e.g. -3 or -2y
Whole number from 0 to 100
1 = first term; leave empty to skip
Enter your values to see the result.
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How to use the Binomial Expansion Calculator
Type the first term a, such as x, 2x or 3x^2.
Type the second term b; use a minus sign for (a − b), for example -3 or -2y.
Enter the power n (a whole number) and, if you need one term only, its term number.
Press Expand to see the full expansion and the steps.
Use the table to check each term, or copy the result.
So (2x + 3)⁴ = 16x⁴ + 96x³ + 216x² + 216x + 81. Check: the coefficients add up to (2 + 3)⁴ = 625.
Binomial coefficients for small powers
n
Coefficients
(a + b)ⁿ
2
1, 2, 1
a² + 2ab + b²
3
1, 3, 3, 1
a³ + 3a²b + 3ab² + b³
4
1, 4, 6, 4, 1
a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴
5
1, 5, 10, 10, 5, 1
a⁵ + 5a⁴b + 10a³b² + 10a²b³ + 5ab⁴ + b⁵
6
1, 6, 15, 20, 15, 6, 1
a⁶ + 6a⁵b + 15a⁴b² + 20a³b³ + 15a²b⁴ + 6ab⁵ + b⁶
Tips
Finding the term with a given power: for (x² − 1/x)⁹ the general term has x to the power 2(9 − k) − k = 18 − 3k. The constant term needs 18 − 3k = 0, so k = 6 and it is term 7. Enter 7 as the term number to see it.
Keep brackets around the whole term when raising it to a power: (2x)³ = 8x³, not 2x³. That is the most common mistake in hand expansions.
Alternating signs appear whenever the second term is negative.
Frequently asked questions
What is the binomial theorem?
The binomial theorem says (a + b)ⁿ = Σ C(n, k)·aⁿ⁻ᵏ·bᵏ for k = 0 to n, where C(n, k) = n! / (k!(n − k)!) is a binomial coefficient. The expansion has n + 1 terms, the powers of a fall from n to 0 while the powers of b rise from 0 to n, and the exponents in each term add up to n.
How do I find a specific term in a binomial expansion?
The (k + 1)th term is Tₖ₊₁ = C(n, k)·aⁿ⁻ᵏ·bᵏ. For example, the 4th term of (x + 2)⁶ uses k = 3: C(6, 3)·x³·2³ = 20·x³·8 = 160x³. Enter the term number in the optional field to get it directly.
How is Pascal's triangle related to binomial expansion?
Row n of Pascal's triangle (starting from row 0) lists the coefficients C(n, 0), C(n, 1), …, C(n, n). Each number is the sum of the two above it. Row 5 is 1, 5, 10, 10, 5, 1, which are the coefficients of (a + b)⁵.
How do I expand (a − b)ⁿ?
Treat it as (a + (−b))ⁿ. The coefficients are the same, but the signs alternate: terms with odd powers of b are negative. For example, (x − 1)³ = x³ − 3x² + 3x − 1. In the calculator, just enter the second term with a minus sign.
What is the middle term?
If n is even there is one middle term, term n/2 + 1, which has the largest binomial coefficient. If n is odd there are two middle terms, terms (n + 1)/2 and (n + 3)/2, with equal binomial coefficients.
What is the sum of the coefficients?
Set every variable to 1. For (2x + 3)⁴ the coefficients add up to (2 + 3)⁴ = 625. The binomial coefficients alone always add up to 2ⁿ.
Can I use a negative or fractional power?
No. For a negative or fractional exponent the binomial series never ends, so there is no finite expansion. This calculator handles whole powers from 0 to 100.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.