Binomial Expansion Calculator

What Is Binomial Expansion?

A binomial is an expression with two terms, such as x + 2 or 3x − 2y. Binomial expansion means multiplying out a power of a binomial, (a + b)ⁿ, into a sum of separate terms. For example (x + 2)³ = (x + 2)(x + 2)(x + 2) = x³ + 6x² + 12x + 8.

Multiplying the brackets one by one works for small powers but quickly becomes long. The Binomial Theorem gives every term of the result directly.

What Is the Binomial Theorem?

For any whole number n ≥ 0: (a + b)ⁿ = C(n,0)aⁿ + C(n,1)aⁿ⁻¹b + C(n,2)aⁿ⁻²b² + … + C(n,n)bⁿ, where C(n,k) = n!/(k!(n − k)!).

There are n + 1 terms. The power of a starts at n and goes down by one each term, the power of b starts at 0 and goes up by one, and the two powers in every term add up to n. The coefficients C(n,k) are the numbers in row n of Pascal's Triangle.

How to Expand a Binomial

Take (x + 2)⁴. Here a = x, b = 2 and n = 4. The coefficients from row 4 of Pascal's Triangle are 1, 4, 6, 4, 1.

Substitute into the theorem: 1·x⁴ + 4·x³·2 + 6·x²·2² + 4·x·2³ + 1·2⁴. Simplify each term: x⁴ + 8x³ + 24x² + 32x + 16.

With a minus sign, include it in b. For (x − 2)³, b = −2, and its powers alternate in sign: x³ + 3x²(−2) + 3x(−2)² + (−2)³ = x³ − 6x² + 12x − 8. With a coefficient, raise the whole term to the power: in (2x + 3)², a² = (2x)² = 4x², giving 4x² + 12x + 9.

How to Find a Specific Term

The (k + 1)th term is Tₖ₊₁ = C(n,k) · aⁿ⁻ᵏ · bᵏ. Because k starts at 0, the rth term uses k = r − 1.

For the 4th term of (x + 2)⁶, k = 3: T₄ = C(6,3) · x³ · 2³ = 20 · 8x³ = 160x³.

To find the coefficient of a particular power, such as x³ in (2x + 3)⁵, set the power of x in the general term equal to 3: 5 − k = 3, so k = 2. Then T₃ = C(5,2)(2x)³(3)² = 10 · 8x³ · 9 = 720x³, and the coefficient is 720.

Binomial Coefficients

The binomial coefficient C(n,k), read “n choose k” and also written ⁿCₖ, counts the ways to choose k items from n. It equals n!/(k!(n − k)!). For example C(10,4) = 10!/(4!·6!) = (10·9·8·7)/(4·3·2·1) = 210.

Useful facts: C(n,0) = C(n,n) = 1; C(n,k) = C(n,n − k), which is why each row of Pascal's Triangle is symmetric; and the coefficients in row n add up to 2ⁿ.

What Is Pascal's Triangle?

Pascal's Triangle starts with 1 at the top. Each row begins and ends with 1, and every other number is the sum of the two numbers above it. Row 4 is 1, 4, 6, 4, 1 and row 5 is 1, 5, 10, 10, 5, 1.

Row n (counting the top row as row 0) lists C(n,0), C(n,1), …, C(n,n), exactly the coefficients of (a + b)ⁿ. For small powers, reading a row of the triangle is the quickest way to get the coefficients.

How to Find the Middle Term

The expansion of (a + b)ⁿ has n + 1 terms. If n is even, n + 1 is odd and there is one middle term, term n/2 + 1 (k = n/2). If n is odd there are two middle terms, terms (n + 1)/2 and (n + 3)/2.

For (x + 2)⁴ the middle term is the 3rd: T₃ = C(4,2)x²·2² = 24x². For (x + 2)⁵ the middle terms are the 3rd and 4th: 40x³ and 80x². When a and b are equal in size, the middle coefficient is the largest in the row.

Binomial Expansion vs FOIL

FOIL (first, outer, inner, last) multiplies two binomials, so it handles (a + b)² = a² + 2ab + b². For higher powers you would have to FOIL repeatedly and collect like terms each time. The Binomial Theorem generalises this to any whole-number power in one step: its coefficients C(n,k) already count how many times each combination appears.

Where Binomial Expansion Is Used

Algebra and calculus: expanding powers, simplifying polynomials, and deriving the power rule for derivatives from (x + h)ⁿ.

Probability and statistics: the binomial distribution P(X = k) = C(n,k)pᵏ(1 − p)ⁿ⁻ᵏ gives the terms of (p + (1 − p))ⁿ, and those terms add up to 1.

Combinatorics: C(n,k) counts subsets, and identities such as Σ C(n,k) = 2ⁿ come from expanding (1 + 1)ⁿ.

Approximation: for small x, (1 + x)ⁿ ≈ 1 + nx, the first two terms of the expansion.

Negative and Fractional Exponents

The finite Binomial Theorem is for whole-number powers n = 0, 1, 2, … With a negative or fractional power, such as (1 + x)⁻² or (1 + x)^(1/2), the expansion never stops: it becomes an infinite series (the generalised binomial series), which is only valid for some values of x (|x| < 1 for (1 + x)ⁿ). This calculator explains that case instead of producing a finite polynomial, which would be wrong.