Binomial Expansion Calculator

THE EXPANSION

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Type a bracket and a power, then press Expand it.

The coefficients, the check, and where it holds

Your bracket

Write it the way the question does: (2x - 3y)^5, (1 + x)^-2, (4 + x)^(1/2). Each side of the bracket is one term, a coefficient may be a fraction or a decimal, and the power may be negative or fractional. Holding ⌫ deletes one character after another.

The bracket and its power

A power that is not a whole number gives an endless series. The bracket then has to be turned into 1 + something first, and the answer only holds over a range of values — both of which are shown.

Tap a box, then use the keys

Every coefficient here is an exact fraction, never a decimal, so (1/2x + 1/3)^4 comes back as fractions rather than as 0.0625. The working shows each binomial coefficient, the powers it multiplies, and the test the page ran on its own answer before printing it.

Examples — tap one and it solves

How the expansion is worked out

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This binomial expansion calculator takes a bracket and a power, written the way a question prints them, and multiplies it out with every coefficient left as an exact fraction.

Type (2x − 3y)^5 and it returns 32x⁵ − 240x⁴y + 720x³y² − 1080x²y³ + 810xy⁴ − 243y⁵. Notice the 32. It is there because the bracket holds 2x, not x, and raising it to the fifth raises the 2 as well.

In a Binomial Expansion the Coefficient Is Not the Whole Story

Most people meet the binomial theorem as a row of numbers — 1, 5, 10, 10, 5, 1 — and assume the numbers are the work. They are the easy part.

The hard part is what each coefficient multiplies. In (2x − 3y)⁵ the third term is 10 × (2x)³ × (−3y)², and every piece of that has to be raised properly: 10 × 8x³ × 9y² = 720x³y². Get the 1, 5, 10, 10, 5, 1 right and the powers wrong, and the answer is still wrong.

A quick check: the two exponents in any term must add to n. In 720x³y² they add to 5.

Why the Numbers Are the Same Ones You Met in Combinations

A binomial expansion of (a + b)⁴ means four brackets multiplied together. Expanding means choosing, from each bracket, whether to take the a or the b, and adding up every way of choosing.

How many ways give exactly two b's? That is the number of ways to pick 2 brackets out of 4, which is 4C2 = 6 — and 6 is the coefficient of a²b². The binomial coefficients are counts. They are not a separate table you have to memorise, and each one is built from factorials in the usual way.

A Binomial Expansion Has No Business Producing Decimals

Ask most calculators for (2.5x + 1.5y)³ and you get 15.625x³ + 28.125x²y + 16.875xy² + 3.375y³. Correct, and useless if the next line of your work needs a fraction.

Feed one a coefficient like a third and the rounding starts at once. Here every coefficient stays a whole-number fraction from the moment your input is read: (x/2 + 1/3)⁴ gives x⁴/16 + x³/6 + x²/6 + 2x/27 + 1/81, and nothing in the working is a decimal.

A Power That Is Not a Whole Number Changes the Rules

(1 + x)⁵ stops after six terms because C(5, 6) is zero. (1 + x)⁻² never stops, because C(−2, k) is never zero. The expansion becomes an infinite series:

1 − 2x + 3x² − 4x³ + 5x⁴ − …

The formula for this case is written for a bracket of the form 1 + something, and for nothing else. That restriction is not decoration. Applying the series to (2 + 3x)⁻¹ as it stands produces an answer that looks perfectly reasonable and is wrong.

Take the Constant Out First

The step that fixes it takes one line, and it is where a great many binomial marks go:

(2 + 3x)⁻¹ = 2⁻¹(1 + 3x/2)⁻¹

Now the bracket is in the right shape. Expand it to get 1 − 3x/2 + 9x²/4 − 27x³/8, then multiply the lot by a half. This page prints that rearrangement as its own step whenever it is needed, and says so.

The power	What you get	What else you must say
a whole number, 0 or more	exactly n + 1 terms, then it stops	nothing
negative whole	an endless series	the range
a fraction	an endless series	the range

Saying Where a Binomial Series Is Valid

A binomial series only equals the bracket where it settles down. For (1 + kx) to a non-whole power that means the size of kx has to stay under 1, which pins x to a range.

For (2 + 3x)⁻¹, rearranged to (1 + 3x/2)⁻¹, the condition is |x| < 2/3. Put x = 1 into the series and the terms grow instead of shrinking; the sum means nothing, and the bracket it was supposed to equal is a perfectly ordinary 1/5.

Exam questions ask for that range explicitly. Every tool I tried prints the terms and stops there, which answers half the question. This page prints the range alongside the terms, because leaving it out is the same mistake in software that it is on paper.

How to Use This Binomial Expansion Calculator

Write the bracket and the power in one line, the way the question has them: (2x - 3y)^5, (1 + x)^-2, (4 + x)^(1/2). The keypad carries the brackets, the caret and the letters, so nothing depends on your phone's keyboard.

Whole powers run up to 100. The tools I compared stop at fifteen, and while nobody sets (x + 1)¹⁰⁰ in an exam, refusing it teaches the wrong lesson about what the theorem can do.

Before anything appears, the expansion is put back through the mathematics. For a finite one the page substitutes x = 1 and x = 2 and checks the total against the bracket worked out directly — a test that never touches the expansion it is testing. For a series with a negative whole power it multiplies the series back by the bracket, which must give 1 and then nothing. The number of checks that passed is printed with the working.

Binomial Expansion Calculator FAQ

What is the binomial expansion of (a + b)^n?

It is the sum of C(n,k) times a to the power n−k times b to the power k, for k running from 0 to n. The powers of a fall as the powers of b rise, and the two always add up to n.

How do you expand (2x − 3y)^5?

Take the coefficients 1, 5, 10, 10, 5, 1 and apply each power to the whole term, the number included. The result is 32x⁵ − 240x⁴y + 720x³y² − 1080x²y³ + 810xy⁴ − 243y⁵.

Why is (2x)³ equal to 8x³ and not 2x³?

The bracket contains the whole term 2x, so cubing it cubes the 2 as well: 2 × 2 × 2 = 8. Forgetting to raise the coefficient is the most common error in the topic.

Can the power be negative or a fraction?

Yes, but the expansion then never ends. The generalised binomial theorem gives an infinite series, and it is written for a bracket of the form 1 plus something, so any other bracket has to be rearranged first.

How do you expand (2 + 3x) to the power −1?

Take the 2 outside first: (2 + 3x)⁻¹ = 2⁻¹(1 + 3x/2)⁻¹. Then expand the bracket as a series to give 1 − 3x/2 + 9x²/4 − 27x³/8, all multiplied by a half. Expanding without rearranging gives a wrong answer that looks plausible.

What is the range of validity of a binomial series?

For (1 + kx) raised to a non-whole power, the series equals the bracket only while the size of kx stays below 1, so |x| < 1/|k|. For (2 + 3x)⁻¹ that works out as |x| < 2/3. Exam questions ask for this and marks are lost by omitting it.

Where do the binomial coefficients come from?

C(n,k) counts how many ways there are to choose the b term from k of the n brackets. That is why the same numbers appear here and in combinations, and why each row of Pascal's triangle is built by adding the pair above.

How many terms does an expansion have?

A whole-number power n gives exactly n + 1 terms. So (x + 1)¹⁰ has eleven, and (x + 1)¹⁰⁰ has a hundred and one. A power that is not a whole number gives infinitely many.

In any binomial expansion: raise the whole term, count the powers to n, and if the exponent is not a whole number, say where your answer lives.