Binomial Theorem Calculator

Binomial Theorem Calculator

Enter the equation and power in the respective fields, and the calculator will determine its binomial expansion with detailed step-by-step calculations.

Binomial Theorem Calculator

The binomial theorem calculator helps you expand binomial expressions quickly and accurately. Manual binomial expansion can be complicated, but this calculator follows the rules of the binomial theorem to provide correct results effortlessly.

What is Binomial Theorem?

In mathematics, a binomial is a polynomial with exactly two terms, separated by a plus or minus sign. The binomial theorem provides a formula to expand powers of a binomial expression in terms of a series of terms. This formula simplifies the expansion of expressions like (x + y)^n without performing repetitive multiplication.

Binomial Theorem Formula

The binomial theorem formula is:

$$ (x + y)^n = \sum_{r=0}^{n} \binom{n}{r} x^{n-r} y^r $$

where:

$ \binom{n}{r} $ is the binomial coefficient calculated as $ \binom{n}{r} = \frac{n!}{r!(n-r)!} $
$ n $ is a non-negative integer
$ x $ and $ y $ are real numbers

Written explicitly, the expansion looks like this:

$$ (a + b)^n = \binom{n}{0} a^n + \binom{n}{1} a^{n-1}b + \binom{n}{2} a^{n-2}b^2 + ... + \binom{n}{n} b^n $$

How to Expand Binomials

Key points to remember when expanding (x+y)^n using the binomial theorem:

The expansion always has n+1 terms.
The sum of the exponents in each term is always n.
Binomial coefficients are $ \binom{n}{0}, \binom{n}{1}, \binom{n}{2}, ... , \binom{n}{n} $.
Coefficients are symmetric: $ \binom{n}{0} = \binom{n}{n}, \binom{n}{1} = \binom{n}{n-1}, \dots $

Binomial Theorem Calculator