Binomial Expansion Calculator

Enter First Term (a): Enter Second Term (b): Enter Power (n):

Result:

📘 Binomial Expansion Calculator – Free Online Tool

Our Binomial Expansion Calculator helps you quickly expand algebraic expressions of the form (a + b)ⁿ using the binomial theorem. Whether you’re a student, teacher, or anyone solving algebra problems, this tool provides instant, accurate binomial expansions without doing long manual calculations.

Binomial expansion is a common concept in algebra, combinatorics, and probability, but expanding expressions by hand can be time-consuming—especially when the exponent is large. This calculator solves that problem with one click.

For more smart maths tools and step-by-step problem solvers, visit our Mathematics Calculators section.

🔍 What Is a Binomial Expansion?

A binomial expansion is the process of expanding an expression raised to a power, such as:

(a + b)ⁿ or (x − y)⁵

Using the Binomial Theorem, the expression is expanded into a sum of terms involving:

binomial coefficients (n choose k)
powers of the first term (a)
powers of the second term (b)

Example:
$2)^3 = x^3 + 6x^2 + 12x + 8$

This calculator automates the entire process and gives you the correct expanded polynomial instantly.

You can also go through - Partial Fraction Decomposition Calculator

🧮 How to Use the Binomial Expansion Calculator

Using our tool is simple and beginner-friendly:

✔ Step 1: Enter the first term (a)

Example input: x, 2x, 3, -4y

✔ Step 2: Enter the second term (b)

Example input: 2, y, -5, 0.5x

✔ Step 3: Enter the exponent (n)

Example inputs: 2, 3, 5, 10

✔ Step 4: Click Calculate Expansion

The calculator will instantly display the full expansion with correct coefficients and simplified terms.

⚙ How the Binomial Expansion Calculator Works

The calculator uses the Binomial Theorem, which states:

$b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{\,n-k} b^k$

Where:

n = power
k = term index
C(n, k) or n choose k = binomial coefficient
a, b = terms in the binomial

The calculator:

Computes the binomial coefficients
Applies the correct powers of the first and second terms
Multiplies and simplifies each term
Joins all parts into a readable polynomial

Everything is automated to avoid manual errors.

⭐ Why Use Our Binomial Expansion Calculator?

✔ Fast and Accurate

Instantly generates expansions even for large exponents that would take minutes by hand.

✔ Supports Variables and Numbers

You can use x, y, z, decimals, negative values, and algebraic expressions.

✔ Perfect for Students & Teachers

Ideal for algebra homework, exam revision, and classroom demonstrations.

✔ Great for Polynomial Problems

Helps in algebra, probability, calculus, and combinatorics.

✔ No Signup, No Fees

Completely free and available online anytime.

✔ Ireland-Friendly Platform

Part of CalcIreland’s growing collection of math and educational calculators designed with students and educators in mind.

❓ FAQs – Binomial Expansion Calculator

1. What is the Binomial Theorem?

The Binomial Theorem provides a formula to expand expressions like (a + b)ⁿ using binomial coefficients, avoiding long multiplications.

2. Can this calculator expand expressions with variables?

Yes. It expands expressions containing variables such as x, y, z, or combinations like 2x, -3y, 0.5x, etc.

3. Does it work with negative exponents?

This version supports non-negative exponents. If you want support for negative or fractional exponents, I can upgrade the tool.

4. Can I enter decimals or fractions?

Yes. The calculator supports integers, decimals, and negative numbers.

5. Is the calculator accurate for large powers?

Yes, the algorithm uses efficient binomial coefficient computation, ensuring accurate results even for high exponents.

Result: