How do you expand a binomial using the binomial theorem?

Take the coefficients from row n of Pascal's triangle, or from C(n, k). Multiply each by a to the power n − k and b to the power k, for k from 0 to n, then simplify and add. For (x + 2)^3 that's x³ + 3·x²·2 + 3·x·4 + 8 = x³ + 6x² + 12x + 8.

How do you use Pascal's triangle to expand a binomial?

Row n of the triangle lists the coefficients of (a + b)^n. Row 4 is 1, 4, 6, 4, 1, so (a + b)^4 = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴. Each number in the triangle is the sum of the two above it.

How many terms are in the expansion of (a + b)^n?

n + 1 terms. (x + 2)^5 has 6 terms, from x⁵ down to the constant 32. Fewer can appear only if some terms cancel or combine, which happens when a and b share a variable.

How do you find a specific term in a binomial expansion?

The (k + 1)th term is C(n, k) · a^(n − k) · b^k. For the 3rd term of (2x − 3)^5, k = 2: C(5, 2) · (2x)³ · (−3)² = 10 · 8x³ · 9 = 720x³. Type the term number in the calculator's second box to get it directly.

What is the sum of the coefficients in a binomial expansion?

Set every variable equal to 1. For (2x − 3)^4 the coefficients add up to (2 − 3)^4 = 1, and for (x + 2)^5 they add to 3^5 = 243. The calculator shows this under the expansion as a quick check.

Can the calculator expand negative or fractional powers?

No. With a whole-number power the expansion stops after n + 1 terms. A negative or fractional power gives the binomial series, which goes on forever and only works for small values, so it needs a different tool.

Binomial Expansion Calculator With Steps

k	C(5, k)	term
0	1	x⁵
1	5	10x⁴
2	10	40x³
3	10	80x²
4	5	80x
5	1	32

Multiplying by itself five times is the kind of job that goes wrong on the fourth line. The binomial theorem skips all of that. It tells you every coefficient in advance and how the powers have to run, so the expansion is a matter of filling in a pattern. This binomial expansion calculator fills it in for you, with exact coefficients and the working shown term by term, and it highlights the row of Pascal’s triangle each answer comes from.

Binomial theorem: (a + b) to the n equals the sum from k = 0 to n of n choose k, times a to the n minus k, times b to the k — The binomial theorem gives every term of (a + b)ⁿ

How to use the binomial expansion calculator

Type the binomial in brackets, then the power. For example (x + 2)^5, (2x - 3)^4 or (x^2 - 2y)^3. Terms can be numbers, letters with powers, fractions like 1/2, or reciprocals like 1/x.
Read the expansion in the dark panel. Under it you’ll see how many terms there are and what the coefficients add up to.
Find one term by typing its position in the second box. Type 3 for the third term and the calculator works it out on its own and highlights it in the table.
Check the triangle. The highlighted row of Pascal’s triangle holds the coefficients before the numbers inside the brackets change them.

The binomial theorem

Read it as a recipe. There are terms, one for each from 0 to . In each term the power of counts down from and the power of counts up from 0, so the two powers always add to . The number in front, , is the binomial coefficient: the number of ways to choose things from , and also entry of row of Pascal’s triangle. The binomial theorem lesson shows why the counting works.

Worked example: (x + 2)⁵

This is the calculator’s opening example.

Step 1 — the pieces. , , .

Step 2 — the coefficients. Row 5 of Pascal’s triangle is 1, 5, 10, 10, 5, 1.

Step 3 — build each term. The powers of fall from 5 to 0 while the powers of 2 rise from 0 to 5:

Step 4 — simplify.

A quick check: put on both sides. The left is , and . The calculator shows this sum on every expansion for exactly this reason.

When the second term is negative

Write the binomial as with negative, and the signs take care of themselves. For , and :

The signs alternate because odd powers of are negative and even powers are positive. Remember to raise the whole term to the power, too: is , not . That one slip causes more wrong answers than anything else in this topic.

Finding one term without expanding everything

Exams love “find the 3rd term” or “find the coefficient of ”, because writing out a whole expansion to the 10th power would take a page. The th term is

The 3rd term of (2x − 3) to the 5th, with k = 2, built as C(5, 2) = 10 from Pascal's triangle, times (2x) cubed = 8x³ because the power of a falls to 5 − k, times (−3) squared = 9 because the power of b rises to k, giving 720x³; a sign check notes that odd powers of the negative b make terms negative — One term of an expansion is just three numbers multiplied

Note the off-by-one: the 3rd term uses , because counting starts at . In the calculator, type (2x - 3)^5 and 3 in the term box to see .

The constant term

For , the general term has from the first part and from the second, so its power of is . That’s zero when , and the constant term is . Choose the (x + 1/x)^6 example to see it sitting in the middle of the expansion.

Expanding with fractions

Fractions follow exactly the same pattern; they just make the arithmetic slower by hand. For , the coefficients are 1, 3, 3, 1 and the powers of fall from 3 to 0:

The calculator keeps every coefficient as an exact fraction, so you’ll see rather than . Type it as (1/2 + x)^3.

Pascal’s triangle and the coefficients

Each row of Pascal’s triangle starts and ends with 1, and every other entry is the sum of the two above it. Row gives the coefficients for power :

n	Pascal row
2	1, 2, 1
3	1, 3, 3, 1
4	1, 4, 6, 4, 1
5	1, 5, 10, 10, 5, 1
6	1, 6, 15, 20, 15, 6, 1

Two things to notice. Each row is symmetric, so you only ever need to work out half of it. And each row adds up to , which is what you get by putting . For large powers the triangle gets unwieldy, and is faster; the calculator handles powers up to 30.

Where binomial expansions are useful

Quick approximations. Write as . The first three terms give , which is . The true value is 1.12616, so three terms already get you four significant figures. Compound interest estimates work the same way.
Probability. The coefficients count outcomes: is the number of ways to get exactly 7 heads in 10 coin flips. The binomial distribution calculator builds on exactly this.
Algebra and calculus. Expanding is the first step in finding the derivative of from the definition.

Common mistakes

Forgetting to raise the coefficient. , not .
Dropping the sign of . In , use every time.
Off-by-one on the term number. The th term uses .
Using row of Pascal’s triangle. The top row is row 0, so uses the row that starts 1, 5.
Expanding as . The middle terms are the whole point; is , not .

Ready to try some by hand? The binomial theorem practice problems cover full expansions, single terms and constant terms, with answers checked as you type.

Binomial Expansion Calculator