Binomial Expansion Calculator
Enter the two terms of a binomial and a power n, and get the full expansion of (a + b)ⁿ with exact coefficients, a table of every term, and the general term formula. Ask for the 3rd term, or for the coefficient of x⁴ or the term without x at all. Terms can be numbers, a letter such as x, or both — 2x, −3y², x⁻¹, 1/2. Free, no sign-up.
A number, a variable, or both: 2x, −3y, x^2, 1/2, x^-1. One letter only.
A whole number from 0 to 100.
From 1 to n + 1, counting from the first term.
Enter p. Use 0 for the term without x.
Results
32x⁵ + 240x⁴ + 720x³ + 1080x² + 810x + 243
Expansion of (2x + 3)⁵
6
Terms after combining
3125
Sum of the coefficients
General term
T(k + 1) = C(5, k) · (2x)^(5 − k) · (3)^k, for k = 0, 1, …, 5
Setting every variable to 1 turns the expansion into a number: (2 + 3)⁵ = 3125.
Term 3
T(3) = C(5, 2) · (2x)³ · (3)² = 10 × 8x³ × 9 = 720x³
Term by term
| k | C(n, k) | a part | b part | Term |
|---|---|---|---|---|
| 0 | 1 | 32x⁵ | 1 | 32x⁵ |
| 1 | 5 | 16x⁴ | 3 | 240x⁴ |
| 2 | 10 | 8x³ | 9 | 720x³ |
| 3 | 10 | 4x² | 27 | 1080x² |
| 4 | 5 | 2x | 81 | 810x |
| 5 | 1 | 1 | 243 | 243 |
How to Use the Binomial Expansion Calculator
- 1. Enter the two terms. Write a and b the way you would on paper:
2x,-3y^2,x^-1,1/2,(3/4)x. A term may use one letter, and the two terms may use the same letter or different ones. Put the sign of b in b: for (x − 2)ⁿ enter-2. - 2. Enter the power. Any whole number from 0 to 100.
- 3. Optionally look for one term. Type a term number to see how that term is built, or a power p to get the coefficient of xᵖ — use 0 for the term without x.
- 4. Read the results. The expansion appears at the top, with the sum of its coefficients, the general term formula, and a table that shows each term's binomial coefficient and its a and b parts.
The Binomial Theorem
(a + b)ⁿ = Σ C(n, k) · aⁿ⁻ᵏ · bᵏ, k = 0, 1, …, n C(n, k) = n! ÷ (k!(n − k)!)
Going from the first term to the last, the power of a falls from n to 0 while the power of b rises from 0 to n, and the two powers always add up to n. The coefficients C(n, k) are the entries of row n of Pascal's triangle: 1, 4, 6, 4, 1 for n = 4. The general term, the (k + 1)th term, is T(k + 1) = C(n, k)·aⁿ⁻ᵏ·bᵏ, and an expansion of (a + b)ⁿ has n + 1 terms before any like terms combine.
Worked Examples
- (2x + 3)⁵. The coefficients are 1, 5, 10, 10, 5, 1, and each term uses (2x)⁵⁻ᵏ and 3ᵏ: 32x⁵ + 240x⁴ + 720x³ + 1080x² + 810x + 243. The 3rd term is C(5, 2)·(2x)³·3² = 10 × 8x³ × 9 = 720x³. As a check, setting x = 1 gives (2 + 3)⁵ = 3125, which is exactly the sum of the coefficients.
- (x − 2)³. With b = −2 the signs alternate: x³ − 6x² + 12x − 8.
- Coefficient of x³ in (1 + x)¹⁰. Enter a = 1, b = x, n = 10, and ask for p = 3. The term is C(10, 3)·1⁷·x³, so the coefficient is 120.
- The term without x in (x² + 1/x)⁶. Enter a =
x^2, b =x^-1. Term k has x to the power 2(6 − k) − k = 12 − 3k. That is 0 when k = 4, so the constant term is C(6, 4) = 15.
| Common mistake | Correct approach |
|---|---|
| Writing (2x)⁵ as 2x⁵ | The whole term is raised to the power: (2x)⁵ = 32x⁵ |
| Losing the sign of b | Treat (x − 2)ⁿ as (x + (−2))ⁿ, so odd powers of b are negative |
| Counting terms from 1 in the formula | Term number r uses k = r − 1, so the first term has k = 0 |
| Expanding (a + b)⁻¹ or (a + b)^(1/2) this way | Those expansions are infinite series, not finite sums — see the Taylor Series Calculator |
Frequently Asked Questions
- How do you expand (a + b)^n?
- Add up C(n, k)·aⁿ⁻ᵏ·bᵏ for k = 0 to n. The coefficients C(n, k) come from row n of Pascal's triangle. For (x + 1)⁴ the coefficients 1, 4, 6, 4, 1 give x⁴ + 4x³ + 6x² + 4x + 1.
- How do I find a specific term of a binomial expansion?
- Use the general term T(k + 1) = C(n, k)·aⁿ⁻ᵏ·bᵏ. The 4th term has k = 3. To find the term containing xᵖ instead, work out the power of x in term k, set it equal to p, solve for k, and then compute that term.
- How do you find the term independent of x?
- Write the power of x in the general term as a function of k, set it to 0, and solve for k. If k is a whole number from 0 to n, the constant term is C(n, k)·aⁿ⁻ᵏ·bᵏ; if not, there is no constant term. Enter p = 0 in the calculator to do this automatically.
- How many terms does a binomial expansion have?
- n + 1 terms for (a + b)ⁿ, from k = 0 to k = n. Some may combine or vanish when a and b contain the same variable. (x + x)ⁿ collapses to a single term, 2ⁿxⁿ.
- What is the sum of the coefficients?
- Set every variable to 1: the sum of the coefficients of (ax + b)ⁿ is (a + b)ⁿ. For (2x + 3)⁵ that is 5⁵ = 3125. It is a quick check on your expansion.
- Can the power be negative or a fraction?
- Not with the finite binomial theorem. (1 + x)⁻¹ and (1 + x)^(1/2) expand into infinite series, valid when |x| < 1, which the Taylor Series Calculator handles. For whole-number powers from 0 to 100 this calculator gives exact finite expansions.
- How does this relate to Pascal's triangle?
- Row n of Pascal's triangle lists C(n, 0), C(n, 1), …, C(n, n), which are exactly the coefficients of (a + b)ⁿ when a and b have coefficient 1. The Pascal's Triangle Generator prints as many rows as you like.