BYTETOOLS

Binomial Expansion Calculator

Expand (ax + b)ⁿ term by term with exact coefficients, find the coefficient of any power of x, and handle fractional or negative exponents as a series.

Expands (ax + b)ⁿ. A whole, non-negative n gives an exact finite expansion; a fractional or negative n switches to the binomial series.

5
Terms
216
Coefficient of x²
Exact integers
Arithmetic

Expansion

(2x + 3)4 = 16x⁴ + 96x³ + 216x² + 216x + 81

Each term is C(n, k)·(ax)n−k·bk, computed with exact big-integer arithmetic so even large exponents keep every digit.

Term by term

Binomial coefficient, power and full coefficient for each term
kC(n, k)power of xcoefficient
01416
14396
262216
341216
41081

What is the Binomial Expansion Calculator?

The ByteTools Binomial Expansion Calculator expands (ax + b)ⁿ using Σ C(n,k)(ax)ⁿ⁻ᵏbᵏ. For a whole, non-negative exponent every coefficient is computed as an exact big integer, so even large powers keep every digit rather than drifting into floating-point approximation.

  • Exact big-integer coefficients for whole, non-negative exponents
  • Finds the coefficient of any requested power of x
  • Term-by-term table showing C(n, k), the power and the coefficient
  • Switches to the binomial series for fractional or negative exponents
  • States the |x| range where the series is genuinely valid
  • Everything computed locally in your browser

How to use the Binomial Expansion Calculator

  1. 1

    Enter a, the coefficient of x, and b, the constant, for your bracket (ax + b).

  2. 2

    Enter the exponent n — whole numbers give an exact finite expansion.

  3. 3

    Type the power of x you want in the 'find the coefficient' box to pick out a single term.

  4. 4

    For a fractional or negative n, set how many series terms to show and read the validity range.

  5. 5

    Click Copy expansion to save the result.

About the Binomial Expansion Calculator

The ByteTools Binomial Expansion Calculator expands (ax + b)ⁿ using Σ C(n,k)(ax)ⁿ⁻ᵏbᵏ. For a whole, non-negative exponent every coefficient is computed as an exact big integer, so even large powers keep every digit rather than drifting into floating-point approximation.

It answers the 'find the coefficient of x⁵' question directly — enter the power you want and the matching term is highlighted in the table and reported on its own. Each row shows the binomial coefficient C(n, k), the power of x and the full coefficient, so you can follow how each term is built.

For a fractional or negative exponent the expansion never terminates, so the tool switches to the binomial series and states the |x| range where it is actually valid. Everything runs in your browser and nothing is uploaded.

Frequently asked questions

What is the binomial theorem?

It says (a + b)ⁿ expands to the sum of C(n,k)·aⁿ⁻ᵏ·bᵏ for k from 0 to n. The coefficients C(n,k) are exactly the entries of Pascal's triangle, which is why the two topics are always taught together.

How do you find a specific term in a binomial expansion?

Pick the k that gives the power of x you want, then compute C(n,k)·(a)ⁿ⁻ᵏ·bᵏ. This calculator does it for you — type the power of x into the box and the matching term is highlighted and reported.

Can you expand a binomial with a fractional or negative power?

Yes, but the expansion becomes an infinite series rather than a finite sum. The coefficients follow n(n−1)…(n−k+1)/k!, and the series only converges for a limited range of x, which this calculator works out and states.

Why does the binomial series only work for small x?

The series for (1 + x)ⁿ with a non-whole n converges only when |x| < 1. Factoring b out of (ax + b)ⁿ rescales that condition, and the tool shows the resulting range. Outside it, the truncated sum does not approximate anything.

How many terms are in the expansion of (ax + b)ⁿ?

For a whole, non-negative n there are exactly n + 1 terms, with powers of x running from n down to 0. For a fractional or negative exponent the expansion is infinite and you choose how many terms to display.

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