Binomial Expansion Calculator

1. Enter the coefficient of x and the constant.
2. Enter the whole-number power.
3. Read the full expansion as a polynomial in x.
4. See the number of terms and the sum of coefficients.

Here is what each result means:

A binomial is a two-term expression, and a binomial expansion writes out that expression raised to a power. The binomial theorem gives every term without multiplying the whole thing out. For (x + 2)^3, the expansion is x^3 + 6x^2 + 12x + 8.

Each term uses a binomial coefficient, one of the numbers in Pascal triangle, times a power of the first term and a power of the second. The powers of the first term count down while the powers of the second count up, and together they always add to n. This pattern appears throughout algebra, probability and calculus.

It applies the binomial theorem, building each term from a binomial coefficient and the two powers.

1. For each term, find the binomial coefficient C(n, k).
2. Multiply by a to the remaining power and b to the k-th power.
3. Attach the matching power of x and add the term to the sum.

The binomial coefficients come from factorials; the factorial calculator shows the counting behind them.

Expand (x + 2)^3.

Calculation: the coefficients from Pascal triangle for power 3 are 1, 3, 3, 1. Multiplying by the powers of 2 gives 1 x^3 + 3 x^2 (2) + 3 x (4) + 1 (8), which simplifies to x^3 + 6x^2 + 12x + 8. There are four terms, one more than the power.

The binomial coefficients for each power are the rows of Pascal triangle.

Each number is the sum of the two above it, and each row starts and ends with 1. These are the coefficients when both terms have a coefficient of 1.

The theorem reaches the same answer as repeated multiplication, but far faster.

For high powers the theorem is the only practical route, and it also lets you pick out a single term without expanding the rest.

The Power N

A higher power gives more terms, always n plus 1 of them, and larger middle coefficients.

The Two Coefficients

The values of a and b scale each term, and a negative b makes the signs alternate.

The Sign of B

When b is negative, the terms alternate between plus and minus down the expansion.

Algebra Homework

Expand a binomial power quickly and check your working.

Finding One Term

Use the pattern to pick out a specific term or coefficient.

Probability

Binomial coefficients count combinations, which underlie the binomial distribution.

1. Forgetting the Coefficients

Every term carries a binomial coefficient. Leaving them out gives the wrong expansion.

2. Wrong Powers

The first term power counts down while the second counts up; together they add to n.

3. Sign Slips with a Negative Term

When b is negative, the signs alternate. Track them carefully.

4. Missing the Power on a Coefficient

When a term has a coefficient like 2x, raise the whole thing, so it becomes a power of 2 as well.

5. Miscounting Terms

A power of n gives n plus 1 terms, not n. Do not drop the first or last.

The expansion is exact; only long coefficients are rounded for display.

It is a rule for expanding a two-term expression raised to a power. Each term is a binomial coefficient times a power of the first term and a power of the second, with the two powers adding to n.

Use the binomial coefficients from Pascal triangle, and pair them with descending powers of the first term and ascending powers of the second. This tool does that for any coefficients and whole-number power.

They are the numbers in each row of Pascal triangle, written C(n, k). They count combinations and set how much each term contributes in the expansion.

Always one more than the power. So a power of 3 gives 4 terms, and a power of 5 gives 6 terms.

The signs alternate down the expansion, plus, minus, plus and so on. For example, (x minus 1) to the fourth is x to the fourth minus 4 x cubed plus 6 x squared minus 4 x plus 1.

It is a triangle of numbers where each entry is the sum of the two above it. Its rows are the binomial coefficients, so it gives the multipliers for a binomial expansion at a glance.

Not as a finite polynomial. Fractional or negative powers give an infinite binomial series, which is a different tool. This calculator handles whole-number powers.

The binomial coefficients count how many ways an outcome can happen, which is the heart of the binomial distribution used for repeated yes-or-no trials.

No. The calculation runs in your browser and nothing you enter is stored or sent anywhere, unless you choose Save, which keeps the result only on this device.

• Binomial theorem (Wikipedia).
• Binomial theorem explained (Maths Is Fun).
• Pascal's triangle (Maths Is Fun).