Binomial Expansion Calculator
Instantly expand any binomial of the form (a + b)ⁿ using the Binomial Theorem. Free, accurate tool showing every coefficient and term for students and educators.
Expand Your Binomial Expression
Enter the terms a and b and choose a power n, and get the full expansion of (a + b)ⁿ instantly, with every binomial coefficient and term shown step by step.
Expansion Details
Enter your binomial to see the full expansion and coefficients
Expansion Results
Full term-by-term binomial expansion
Pascal’s Triangle
Standard binomial coefficients for powers 0 through 8, taken directly from Pascal’s Triangle, to help you quickly verify your expansions.
| n | Coefficients C(n, k) | ||||||||
|---|---|---|---|---|---|---|---|---|---|
Binomial Expansion FAQ
Everything you need to know about the Binomial Theorem, coefficients, and expanding expressions.
The Binomial Theorem describes how to expand an expression of the form (a+b) raised to a positive integer power n. It states that (a+b)ⁿ equals the sum of C(n,k) · an−k · bk for k = 0 to n, where C(n,k) is the binomial coefficient.
Binomial coefficients are calculated using the combination formula C(n,k) = n! / (k! × (n−k)!). They can also be read directly from the corresponding row of Pascal’s Triangle, where each row gives the coefficients for that power.
Pascal’s Triangle is a triangular array of numbers where each entry is the sum of the two entries directly above it. Row n of the triangle gives the binomial coefficients for the expansion of (a+b)ⁿ, making it a quick way to check your work.
The expansion of (a+b)ⁿ always has exactly n+1 terms, with exponents on ‘a’ decreasing from n down to 0 and exponents on ‘b’ increasing from 0 up to n, so the total exponent in every term always sums to n.