What does this binomial expansion calculator actually compute?

It applies the binomial theorem, which says (a + b)ⁿ equals the sum over k from 0 to n of C(n,k)·a^(n-k)·b^k. For each k it shows the coefficient C(n,k), the power on a, and the power on b. So (a + b)² expands to a² + 2ab + b² and (a + b)³ to a³ + 3a²b + 3ab² + b³. You read the whole expansion at once instead of looking up one coefficient.

How do I expand something like (2x + 3)⁴ with real numbers?

Turn on numeric mode, set the exponent to 4, the first coefficient to 2 and the second to 3, and keep the variable as x. The tool substitutes a = 2 and b = 3 into every term, so it returns 16x⁴ + 96x³ + 216x² + 216x + 81. Each coefficient is C(4,k)·2^(4-k)·3^k worked out exactly, for example the middle term is C(4,2)·2²·3² = 6·4·9 = 216.

What is the binomial coefficient C(n,k)?

C(n,k), read as n choose k, counts how many ways to pick k items from n, and it is exactly the coefficient of the k-th term in the expansion. It equals n! divided by k!·(n-k)!. For (a + b)⁴ the coefficients are C(4,0) through C(4,4), which are 1, 4, 6, 4, 1. This tool computes C(n,k) with BigInt so even C(100,50), a thirty-digit number, is exact.

How is this related to Pascal's triangle?

Row n of Pascal's triangle is exactly the list of coefficients in the expansion of (a + b)ⁿ. Row 4 reads 1 4 6 4 1, which is why (a + b)⁴ has those five coefficients. Each entry is the sum of the two above it, and also equals C(n,k). If you want the triangle itself, our Pascal's triangle tool draws it; here you get the full algebraic terms with their powers attached.

How do I find just the k-th term without the whole expansion?

Use the general-term box. The (k+1)th term of (a + b)ⁿ is T = C(n,k)·a^(n-k)·b^k, counting from k = 0, so the first term is k = 0 and the last is k = n. Enter your k and the tool returns that single term. For (a + b)⁷ with k = 2 it gives C(7,2)·a⁵·b² = 21a⁵b². This is the standard way exam questions ask for a particular term or a specific coefficient.

Why use BigInt instead of ordinary numbers?

Binomial coefficients grow past the safe-integer limit of JavaScript (about 9 quadrillion) surprisingly fast: C(60,30) is already 17 digits. Once you substitute numeric coefficients it is worse, because a^(n-k)·b^k for large n is enormous. Ordinary floating-point numbers would round those values and quietly give wrong digits. Every figure here is a BigInt, so each coefficient and each substituted term is exact to the last digit.

Check homework on the binomial theorem

You expanded (a + b)⁵ by hand and got 1, 5, 10, 10, 5, 1 for the coefficients but want to be sure before you hand it in. Type n = 5 and the tool lists every term with its powers, so you can match a⁵, 5a⁴b, 10a³b², 10a²b³, 5ab⁴, b⁵ against your own work line by line. When numeric substitution is needed, switch it on and confirm the number in front of each power.

Find one specific term in an exam-style question

A typical question asks for the coefficient of x³ in (2x + 3)⁵ rather than the whole expansion. Turn on numeric mode with a = 2, b = 3, then use the general-term box: the x³ power sits at k where the a power is 3, and the tool returns that single term with its exact numeric coefficient. No need to expand all six terms by hand.

Teach the link between Pascal's triangle and (a + b)ⁿ

Showing a class why row 4 of Pascal's triangle is 1 4 6 4 1? Expand (a + b)⁴ here and the coefficients line up with the triangle row, each one a C(4,k). Project the symbolic expansion next to the triangle and students see the same numbers appear in two places, which makes the identity stick better than a formula on its own.

Generate exact coefficients for a series or approximation

Working on a polynomial approximation and need the first several coefficients of (1 + x)ⁿ without rounding error? Set numeric mode with a = 1 and your b, read the coefficient list straight off, and copy it. Because everything is BigInt, the large central coefficients stay exact rather than drifting the way floating-point would.

Binomial Expansion Calculator — Expand (a + b)ⁿ Term by Term

Expand (a + b)ⁿ term by term with exact BigInt coefficients, substitute numbers like (2x + 3)⁴, and pull out any single term — browser-only

Runs locally
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Best for Getting a realistic range before a purchase, plan, workout, or schedule decision.

Expand (a + b)ⁿ into its terms. Every coefficient is a binomial coefficient C(n,k), exact via BigInt, so even (a+b)^100 comes out with every digit right. Switch on numeric mode to substitute real coefficients — (2x + 3)^4 expands with the actual number in front of each power.

Exponent n (0–100)Numeric coefficients (substitute a, b)

Expansion

a^4 + 4·a^3·b + 6·a^2·b^2 + 4·a·b^3 + b^4

Coefficients: 1, 4, 6, 4, 1

Setting a = b = 1 makes every term 1, so the coefficients sum to 2^4 — the row-sum identity.

Find one term (general term T₍k+1₎)

k (0-based, so k=0 is the 1st term)

Term number 2 (k = 1) =4·a^3·b

What this tool does

A binomial expansion calculator that writes out every term of (a + b)ⁿ, not just a single coefficient. Enter an exponent n from 0 to 100 and the tool lists each term in the form C(n,k)·a^(n-k)·b^k: the binomial coefficient, the falling power on a, and the rising power on b. Every coefficient is computed with BigInt, so the whole row stays exact even when it runs to thirty digits — C(100,50) comes out with every digit intact, where a plain calculator silently rounds. Flip on numeric mode to substitute real coefficients and a variable: (2x + 3)⁴ expands with the actual number in front of each power of x, and (x − 1)⁵ alternates sign the way the binomial theorem says it should. There is also a general-term box: give it k and it returns just the (k+1)th term without printing the whole expansion. One-click copy and a shareable URL that reopens your exact exponent and coefficients. Everything runs in your browser, nothing is uploaded.

Tool details

Input: Files + Numbers; The page exposes text boxes, numeric controls, file pickers, or structured inputs depending on the tool.
Output: Live result + Copy; The result area focuses on usable output, with copy, download, or preview actions when supported.
Privacy: Browser-side processing; The main tool logic does not call an external API, so inputs normally stay in the current tab.
Save / share: Shareable URL state; Key settings are encoded in the URL so another person can reopen the same setup.
Performance budget: Initial JS <= 9 KB; No WASM budget is declared, keeping the tool quick to open on mobile.
Best fit: Calculator · Student; Category and role tags drive related tools, internal links, and quick fit checks.

How to use

1. Input

Paste or drop your content into the tool panel.
2. Process

Click the button. All processing is local in your browser.
3. Copy / Download

Copy the result or download to disk in one click.

How Binomial Expansion Calculator fits into your work

Use it for fast estimates, comparisons, and planning numbers before you make the final call.

Calculation jobs

Getting a realistic range before a purchase, plan, workout, or schedule decision.
Comparing scenarios by changing one input at a time.
Turning rough assumptions into a number you can discuss.

Calculation checks

Double-check units, dates, rates, and rounding assumptions.
Treat health, finance, tax, and legal outputs as planning aids, not professional advice.
Save the inputs that produced an important result so you can reproduce it later.

Good next steps

These links move the current task into a more complete workflow.

Real-world use cases

Check homework on the binomial theorem
You expanded (a + b)⁵ by hand and got 1, 5, 10, 10, 5, 1 for the coefficients but want to be sure before you hand it in. Type n = 5 and the tool lists every term with its powers, so you can match a⁵, 5a⁴b, 10a³b², 10a²b³, 5ab⁴, b⁵ against your own work line by line. When numeric substitution is needed, switch it on and confirm the number in front of each power.
Find one specific term in an exam-style question
A typical question asks for the coefficient of x³ in (2x + 3)⁵ rather than the whole expansion. Turn on numeric mode with a = 2, b = 3, then use the general-term box: the x³ power sits at k where the a power is 3, and the tool returns that single term with its exact numeric coefficient. No need to expand all six terms by hand.
Teach the link between Pascal's triangle and (a + b)ⁿ
Showing a class why row 4 of Pascal's triangle is 1 4 6 4 1? Expand (a + b)⁴ here and the coefficients line up with the triangle row, each one a C(4,k). Project the symbolic expansion next to the triangle and students see the same numbers appear in two places, which makes the identity stick better than a formula on its own.
Generate exact coefficients for a series or approximation
Working on a polynomial approximation and need the first several coefficients of (1 + x)ⁿ without rounding error? Set numeric mode with a = 1 and your b, read the coefficient list straight off, and copy it. Because everything is BigInt, the large central coefficients stay exact rather than drifting the way floating-point would.

Common pitfalls

Counting terms from 1 instead of 0. The general term is T = C(n,k)·a^(n-k)·b^k with k starting at 0, so the very first term is k = 0, not k = 1. If a question asks for the 4th term, that is k = 3. The tool labels both the term number and k so you do not slip a position.
Forgetting the falling and rising powers. In (a + b)ⁿ the power on a goes down from n to 0 while the power on b goes up from 0 to n, and the two always add to n. A common slip is keeping a^n in every term; check that each term's a-power and b-power sum to n.
Dropping the sign when one term is negative. For (x − 1)ⁿ the second coefficient is −1, so the signs alternate plus and minus across the expansion. Enter b = −1 in numeric mode rather than 1, otherwise every term comes out positive and the answer is wrong by sign on the odd terms.

Privacy

Every step — the binomial coefficients, the symbolic terms, the numeric substitution and the single-term lookup — is plain JavaScript that runs in your browser tab. No exponent, coefficient or result ever leaves the page, and nothing you type is logged. The one caveat: the shareable link encodes your exponent and coefficients in the query string, so a link pasted into chat records those values in the recipient server's access log. That is fine for math homework; just know the numbers travel with the URL.

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Binomial Expansion Calculator — Expand (a + b)ⁿ Term by Term

What this tool does

Tool details

How to use

1. Input

2. Process

3. Copy / Download

How Binomial Expansion Calculator fits into your work

Calculation jobs

Calculation checks

Good next steps

Real-world use cases

Check homework on the binomial theorem

Find one specific term in an exam-style question

Teach the link between Pascal's triangle and (a + b)ⁿ

Generate exact coefficients for a series or approximation

Common pitfalls

Privacy

FAQ

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