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Binomial Calculator

Enter a, b, and a nonnegative exponent to get the expanded polynomial and any selected binomial-theorem coefficient.

Expand a Binomial

Enter a, b, and n to expand (ax + b)^n; optionally enter k to inspect one theorem coefficient.

The expanded polynomial, term count, and optional k coefficient will appear here.

How to Use Binomial Calculator

1

Enter the Two Coefficients

a multiplies x, while b is the constant in (ax + b).

2

Enter the Exponent

Use a whole number from 0 through 20.

3

Inspect the Expansion

Read terms in descending powers of x, and optionally enter k to inspect one numeric coefficient.

Binomial Calculator Features

  • Expand (ax + b)^n
  • Positive and negative coefficients
  • Optional k coefficient lookup
  • Exponents from 0 through 20

Common Use Cases

Check an algebra expansion

Compare a hand-expanded binomial with every coefficient and power generated by the theorem.

Inspect one theorem term

Enter k to see the numeric coefficient multiplying x^(n−k) without rebuilding the entire term manually.

Limits and Important Notes

Result type: Exact calculation

Calculation Details

  • Generate the binomial coefficient C(n,k) for each k.
  • Multiply by a^(n−k) and b^k to obtain the coefficient of x^(n−k).
  • Join the nonzero terms in descending powers of x.

Formula

(ax+b)^n = Σ[k=0..n] C(n,k)(ax)^(n−k)b^k.

Rounding

Coefficients are calculated numerically and displayed with up to twelve significant digits; very large values may use scientific notation.

Default Assumptions

  • The variable is x.
  • The exponent n is a nonnegative integer no greater than 20.
  • a and b are finite real numeric coefficients between −1,000,000 and 1,000,000.

Verified Example

Inputs: a = 2, b = 3, n = 3, and optional k = 1.

Calculation: Coefficients are C(3,k)×2^(3−k)×3^k for k = 0, 1, 2, 3.

Result: (2x + 3)^3 = 8x^3 + 36x^2 + 54x + 27; the k = 1 coefficient is 36.

Sources and Version

Version: Finite binomial theorem for nonnegative integer n

Last verified: 2026-08-15

Update responsibility: OnlineTool.me

  • The calculator handles one variable and nonnegative integer exponents only.
  • It does not factor polynomials, solve equations, or expand products of several unrelated expressions.
  • Floating-point display can round coefficients when extremely large or when decimal coefficients are entered.

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Frequently Asked Questions