Binomial Theorem Calculator
This binomial theorem calculator expands (a + b)^n term by term. For (2x - 3)^4 it multiplies each coefficient from row 4, 1 4 6 4 1, by the matching powers of 2x and -3.
Expansion
- Coefficients from row 4 The binomial coefficients C(4, k) for k = 0 to 4.
- 1, 4, 6, 4, 1
- General term
- C(4, k) (2x)^(4 - k) (-3)^k
- Number of terms One more than the power: 4 + 1.
- 5
- Sum of the coefficients Set every variable to 1: (2 + (-3))^4 = 1. A quick check on any expansion.
- 1
| k | C(4, k) | Term |
|---|---|---|
| 0 | 1 | 16x^4 |
| 1 | 4 | -96x^3 |
| 2 | 6 | 216x^2 |
| 3 | 4 | -216x |
| 4 | 1 | 81 |
How the Pascal's Triangle works
The binomial theorem says how to expand a power of a sum without multiplying it out:
(a + b)^n = C(n, 0)a^n + C(n, 1)a^(n-1)b + C(n, 2)a^(n-2)b^2 + ... + C(n, n)b^n
The coefficients C(n, k) are the entries of row n of Pascal's triangle, and in each term the power of a falls by one while the power of b rises by one, so the two powers always add to n.
A worked example. For (2x - 3)^4, a is 2x, b is -3 and n is 4. Row 4 is 1 4 6 4 1. The five terms are:
k = 0: 1 x (2x)^4 = 16x^4
k = 1: 4 x (2x)^3 x (-3) = 4 x 8x^3 x (-3) = -96x^3
k = 2: 6 x (2x)^2 x (-3)^2 = 6 x 4x^2 x 9 = 216x^2
k = 3: 4 x (2x) x (-3)^3 = 4 x 2x x (-27) = -216x
k = 4: 1 x (-3)^4 = 81
So (2x - 3)^4 = 16x^4 - 96x^3 + 216x^2 - 216x + 81. The signs alternate because the odd powers of -3 are negative.
A check that catches most mistakes. Set every variable to 1. The left side becomes (2 - 3)^4 = 1, and the coefficients on the right add up to 16 - 96 + 216 - 216 + 81 = 1. If the two disagree, a term is wrong.
Entering the terms. Like any binomial expansion calculator, this one needs the two terms and the power. Each term is one number, one letter, or a number times a letter with an optional power: x, 2x, -y, x^2, 3 or 1/2y. A subtraction is a negative second term, so (x - 2y)^5 is a = x and b = -2y. Fractions stay exact: (x + 1/2)^3 expands to x^3 + (3/2)x^2 + (3/4)x + 1/8.
Finding one term. The term with b^k is C(n, k) a^(n-k) b^k, so the x^2 term of (2x - 3)^4 is k = 2: 6 x 4x^2 x 9 = 216x^2.
Pascal's Triangle FAQ
What is the binomial theorem?
(a + b)^n equals the sum of C(n, k) a^(n-k) b^k for k from 0 to n, where C(n, k) = n! / (k!(n - k)!) is entry k of row n of Pascal's triangle.
How do I expand (2x - 3)^4?
Use row 4 of Pascal's triangle, 1 4 6 4 1, with a = 2x and b = -3: 16x^4 - 96x^3 + 216x^2 - 216x + 81.
How many terms does (a + b)^n have?
n + 1, one for each k from 0 to n, unless some terms combine. (a + b)^4 has five.
How do I find a specific term of a binomial expansion?
The term containing b^k is C(n, k) a^(n-k) b^k. For the x^2 term of (2x - 3)^4, take k = 2: 6 x (2x)^2 x (-3)^2 = 216x^2.
Why do the signs alternate?
Because the second term is negative. (-3)^k is positive for even k and negative for odd k, so the terms go plus, minus, plus. If both terms are positive, every coefficient is positive.