Binomial Expansion Calculator
Expand any expression of the form (a + b)^n using the binomial theorem. Enter values for a, b, and the exponent n (up to 12), and see every term with its binomial coefficient. For example, (2 + 3)^4 expands to 16 + 96 + 216 + 216 + 81 = 625.
The Binomial Theorem
The binomial theorem states that:
(a + b)^n = Σ C(n, k) x a^(n-k) x b^k for k = 0 to n
where C(n, k) = n! / (k! x (n-k)!) is the binomial coefficient, also written as "n choose k."
Pascal's Triangle
The binomial coefficients form Pascal's triangle. Each row gives the coefficients for a particular power:
- n=0: 1
- n=1: 1, 1
- n=2: 1, 2, 1
- n=3: 1, 3, 3, 1
- n=4: 1, 4, 6, 4, 1
Each number is the sum of the two numbers directly above it.
Worked Example: (x + 2)^3
Using C(3,0)=1, C(3,1)=3, C(3,2)=3, C(3,3)=1:
(x + 2)^3 = 1 x x^3 + 3 x x^2 x 2 + 3 x x x 4 + 1 x 8 = x^3 + 6x^2 + 12x + 8
Key Properties
- The expansion always has n + 1 terms
- Coefficients are symmetric: C(n, k) = C(n, n-k)
- The sum of all coefficients equals 2^n (set a = b = 1)