🔺 Binomial Expansion Calculator
(2x − 3)⁴ = 16x⁴ − 96x³ + 216x² − 216x + 81 — expanded with exact BigInt coefficients, plus the k-th term on demand and Pascal's row for n.
A and B accept integers, decimals and fractions; negative B: type the minus sign. Full expansion shown for n ≤ 12.
(2x − 3)⁴ = 16x⁴ − 96x³ + 216x² − 216x + 81
Term finder — term with x⁴⁻ᵏ (choose k = 0…4)
Pascal's triangle row 4 (the binomial coefficients)
1 4 6 4 1
Coefficients are exact BigInt/rational values via the binomial theorem — (2x − 3)⁴ expands with every sign and coefficient right. Runs locally.
How the binomial expansion calculator works
The binomial theorem says (A + B)ⁿ = Σ C(n,k)·Aⁿ⁻ᵏ·Bᵏ, and the tool computes each term exactly: binomial coefficients as big integers, powers of your (possibly fractional, possibly negative) A and B as exact rationals, signs falling out naturally from the arithmetic rather than from error-prone alternation rules. Full expansions display up to n = 12; beyond that the term finder takes over — give k and get the exact term C(n,k)·Aⁿ⁻ᵏ·Bᵏ·xⁿ⁻ᵏ for any n up to 200, which is how "find the coefficient of x⁷ in (…)²⁰" problems are actually solved. Pascal's triangle row n is shown alongside for n ≤ 30.
Sign errors are the classic binomial-expansion failure — with B negative, terms alternate only when they should, and hand-alternating "+ − + −" breaks the moment A is also negative or a term's power is even. Exact arithmetic sidesteps the rule entirely: each term's sign is computed, not remembered. The term finder mirrors exam technique: nobody expands (x + 2)²⁰ fully to find one coefficient — you go straight to k with the formula.
Frequently asked questions
What is the binomial theorem?
(A + B)ⁿ expands to the sum of C(n,k)·Aⁿ⁻ᵏ·Bᵏ for k = 0…n, where C(n,k) are the binomial coefficients — row n of Pascal's triangle. The tool applies it with your A, B and n, exactly.
How do I find one specific term without expanding everything?
Use the general term: term k has coefficient C(n,k)·Aⁿ⁻ᵏ·Bᵏ and power xⁿ⁻ᵏ. To find the x⁷ term of a degree-20 expansion, set k = 13 — the term finder does this for any k.
Why do signs alternate when B is negative?
Because Bᵏ flips sign with each k: (−3)¹ = −3, (−3)² = +9, and so on. The tool computes each Bᵏ exactly, so the signs are always right — including the tricky cases where A is negative too.
What is Pascal's triangle's role?
Row n lists the coefficients C(n,0)…C(n,n) — each entry the sum of the two above it. For (A+B)⁴ the row is 1 4 6 4 1. The tool prints the row for your n (up to 30) as a cross-check.
Do fractional coefficients work?
Yes — (x/2 + 3)⁵ expands with exact fractional coefficients (rationals, not decimals), and n can go to 200 via the term finder with full BigInt precision.