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Binomial Expansion Calculator (Binomial Theorem and Pascal’s Triangle)

Expand (ax + b)ⁿ or (ax + by)ⁿ with the binomial theorem: every term with its binomial coefficient, the Pascal’s triangle row, the middle term and the sum of coefficients.

a (coefficient of x)

b (second term)

Power n

Form

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How it works

(ax + b)ⁿ = Σₖ C(n, k)·(ax)ⁿ⁻ᵏ·bᵏ, k = 0 … n · C(n, k) = n! / (k!(n − k)!) = row n of Pascal’s triangle · sum of coefficients = (a + b)ⁿ

The binomial theorem expands a power of a two-term expression without multiplying it out repeatedly. Term k is C(n, k)·(ax)ⁿ⁻ᵏ·bᵏ, where the binomial coefficient C(n, k) is the k-th entry of row n of Pascal’s triangle (each entry is the sum of the two above it). Powers of the first term fall from n to 0 while powers of the second rise from 0 to n, so there are always n + 1 terms. Substituting x = 1 shows the coefficients add up to (a + b)ⁿ — a quick check on any expansion.

Worked example

Expand (2x − 3)⁴. Row 4 of Pascal’s triangle is 1, 4, 6, 4, 1. The terms are 1·(2x)⁴ = 16x⁴, 4·(2x)³·(−3) = −96x³, 6·(2x)²·9 = 216x², 4·(2x)·(−27) = −216x and 1·81 = 81, so (2x − 3)⁴ = 16x⁴ − 96x³ + 216x² − 216x + 81. Check: the coefficients sum to (2 − 3)⁴ = 1.

Sources: Binomial theorem — Wikipedia · Pascal’s triangle — Wikipedia

Frequently asked questions

How do I find a specific term without expanding everything?

Use the general term T(k+1) = C(n, k)·(ax)ⁿ⁻ᵏ·bᵏ. For the x² term of (2x − 3)⁴ you need n − k = 2, so k = 2: C(4, 2)·(2x)²·(−3)² = 6·4x²·9 = 216x². The table lists every term with its k.

Why do the signs alternate?

When the second term is negative, odd powers of it are negative and even powers positive, so the terms alternate +, −, +, −.

How is Pascal’s triangle built?

Each row starts and ends with 1, and every inner entry is the sum of the two entries above it: 1; 1 1; 1 2 1; 1 3 3 1; 1 4 6 4 1 …. Row n gives the coefficients of (x + y)ⁿ.

Can I expand (x + y)ⁿ?

Yes — choose the form (ax + by)ⁿ and set a = 1, b = 1. The expansion then shows both variables with their powers.

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This calculator is for educational purposes. Double-check important results before acting on them.