Binomial Expansion Calculator

What Is a Binomial Expansion Calculator?

A binomial expansion calculator is a free online tool that expands any expression of the form (A + B)^n or (A - B)^n using the Binomial Theorem, instantly and accurately. Instead of manually multiplying a binomial by itself n times — which becomes extremely tedious and error-prone as n grows — this calculator applies the binomial theorem formula directly, computing every binomial coefficient, every power of A and B, and the final combined term in one step.

This tool works for any binomial you type in: numeric binomials like (2 + 3)^5, algebraic binomials like (x + y)^6, mixed expressions like (2x - 3y)^4, or single-variable binomials like (x + 2)^7. Just enter the coefficients, the variable names, the sign between the two terms, and the power n, and the calculator returns the complete expansion along with a full step-by-step solution and a visual diagram.

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Google favors pages that directly satisfy what a searcher is looking for. Someone who searches for a binomial expansion calculator usually wants three things at once: an instant, correct answer; the formula and reasoning behind that answer; and a visual way to check their own work. A page that combines a working calculator, the binomial theorem formula, clear step-by-step working, and a labeled diagram answers all three intents on a single page, which is exactly what search engines reward with better rankings.

This page is built around the keywords people actually search for, including binomial expansion calculator, binomial theorem calculator, expand binomial calculator, (a+b)^n calculator, Pascal's triangle calculator, binomial coefficient calculator, expand (x+y)^n, general term binomial expansion, and nCr calculator. These terms are woven naturally through the explanations below so the page reads well for both students and search engines.

The Binomial Theorem Formula

The binomial theorem states that for any non-negative integer n, the expansion of (A + B)^n can be written as a sum of n + 1 terms. Each term uses a binomial coefficient C(n,k), also written as nCk or 'n choose k', multiplied by decreasing powers of A and increasing powers of B. Formally, (A + B)^n = Σ (from k = 0 to n) of C(n,k) · A^(n-k) · B^k. When the second term is negative, as in (A - B)^n, the signs of the terms simply alternate because B is effectively replaced with -B.

The binomial coefficient itself is calculated with factorials: C(n,k) = n! / (k! · (n-k)!). These coefficients are exactly the numbers that appear in Pascal's Triangle, where each row corresponds to a power n, and each entry in that row is one coefficient of the expansion. This connection is why Pascal's triangle calculator and binomial coefficient calculator searches so often lead to the same kind of tool.

• General formula: (A + B)^n = Σ C(n,k) · A^(n-k) · B^k
• Binomial coefficient: C(n,k) = n! / (k!(n-k)!)
• Number of terms in the expansion: always n + 1
• Powers of A decrease from n down to 0
• Powers of B increase from 0 up to n
• Sum of all coefficients in the expansion: 2^n

How to Use the Binomial Expansion Calculator

Using the calculator takes only a few seconds. Enter the coefficient and variable name for the first term (A), choose the sign between the two terms, enter the coefficient and variable name for the second term (B), and select the power n. The result updates instantly, showing the fully expanded expression, a colour-coded diagram of every individual term, Pascal's triangle for that row, and a complete step-by-step solution underneath.

The diagram is especially useful because every term is displayed as its own card, showing the value of k, the binomial coefficient C(n,k), the exact powers of A and B used, and the final calculated value of that term — all in one glance. This makes it far easier to understand than a plain formula, especially for students who are seeing the binomial theorem for the first time.

• Enter coefficient A and its variable (or leave the variable blank for a plain number)
• Choose + or − for the sign between the two terms
• Enter coefficient B and its variable
• Select the power n
• Read the expanded result, diagram, and step-by-step solution

Worked Example: Expanding (x + 2)^4

Take the binomial (x + 2)^4, so A = x, B = 2, and n = 4. Row 4 of Pascal's Triangle gives the coefficients 1, 4, 6, 4, 1. Applying the binomial theorem term by term: the first term is 1 · x^4 = x^4; the second is 4 · x^3 · 2 = 8x^3; the third is 6 · x^2 · 4 = 24x^2; the fourth is 4 · x · 8 = 32x; and the fifth is 1 · 16 = 16.

Adding every term together gives the final expansion: x^4 + 8x^3 + 24x^2 + 32x + 16. The calculator's diagram displays each of these five terms as a separate coloured card, so you can trace exactly how the coefficient and powers combine to build each part of the answer, and you can double-check the arithmetic against the same values shown in Pascal's triangle.

Worked Example: Expanding (2x − y)^3

Now try (2x - y)^3, where A = 2x, B = y with a minus sign, and n = 3. Row 3 of Pascal's Triangle gives coefficients 1, 3, 3, 1. Because the sign is negative, the terms alternate between positive and negative: the first term is 1 · (2x)^3 = 8x^3; the second is 3 · (2x)^2 · (-y) = -12x^2y; the third is 3 · (2x) · (-y)^2 = 6xy^2; and the fourth is 1 · (-y)^3 = -y^3.

The complete expansion is 8x^3 - 12x^2y + 6xy^2 - y^3. This example shows why the sign toggle matters so much: forgetting that B is negative is one of the most common mistakes when expanding binomials by hand, and the calculator handles it automatically for every term.

Understanding Pascal's Triangle and Binomial Coefficients

Pascal's Triangle is a triangular arrangement of numbers where each row represents the binomial coefficients for a specific power n, and each number is the sum of the two numbers directly above it. Row 0 is just 1; row 1 is 1, 1; row 2 is 1, 2, 1; row 3 is 1, 3, 3, 1; and so on. These are exactly the same values produced by the combination formula C(n,k), which is why Pascal's triangle and the binomial theorem are so closely linked.

This calculator draws the full triangle up to the row you need and highlights the exact row used in your expansion, so you can visually confirm where each coefficient comes from instead of trusting the formula blindly.

The General Term (r + 1th Term) Formula

Sometimes you don't need the entire expansion — just one specific term, such as the 5th term of (x + 3)^10. This is called the general term, written T(r+1) = C(n,r) · A^(n-r) · B^r, where r starts at 0 for the first term. The general term finder in this calculator lets you pick any term number directly and see its coefficient, powers, and final value without expanding the whole binomial.

This feature is particularly useful for exam-style questions that ask for 'the coefficient of x^5 in the expansion of...' or 'the middle term of...', since you can jump straight to the relevant term instead of computing every single one.

Common Mistakes in Binomial Expansion

A frequent mistake is forgetting to apply the sign to every occurrence of B when the binomial has a minus sign — the sign must be raised to the same power as B itself, not just attached once at the end. Another common error is mismatching the exponents, since the power of A must decrease by exactly one for every step that the power of B increases by one, and the two must always add up to n.

Students also sometimes miscalculate the binomial coefficient itself, especially for larger values of n where factorials grow quickly. Using Pascal's Triangle as a visual cross-check, exactly as this calculator does, is one of the most reliable ways to avoid that mistake.

Applications of Binomial Expansion

Binomial expansion is a foundational tool in algebra, probability, statistics, calculus, and engineering. In probability theory, the binomial coefficients directly describe the number of ways an event can occur, which is why the binomial theorem underlies the binomial probability distribution. In calculus, binomial expansions are used to approximate functions and derive series expansions. In engineering and physics, binomial approximations simplify complex expressions when one term is much smaller than another.

Because the binomial theorem shows up in so many contexts, people search for it in different ways — binomial expansion calculator, expand (a+b)^n, binomial theorem solver, Pascal's triangle generator, or general term of binomial expansion. This calculator is built to answer all of these related searches with one flexible, easy-to-use tool.