Binomial Theorem Calculator
Binomial Expansion Calculator expressions using combinatorial coefficients. Enter two terms and a nonnegative integer power to generate simplified polynomial results instantly.
Equation Preview
Helping Notes
Only three inputs are required. The expansion uses C(n,k) = n! / (k!(n−k)!) for k = 0 … n.
Expressions are simplified term-by-term. Use consistent variable names if you want like terms to combine naturally.
Results
Inputs Used
Coefficient Row C(n,k)
Expanded Sum
Term-by-Term Details
General Term
What Is a Binomial Theorem Calculator?
The Binomial Theorem Calculator expands powers of a binomial such as into a sum of terms with precise coefficients. It uses the binomial theorem to write the expansion in closed form, shows each coefficient, returns both symbolic and numeric versions, and highlights structure like symmetry and the number of terms. You can input variables, parameters, or numbers, and the calculator will display the series, identify a requested coefficient, and simplify expressions automatically. This is useful for algebra practice, combinatorics, series manipulation, probability moment expansions, and quick checks when coding or typesetting.
About the Binomial Theorem Calculator
The theorem states for integers . The binomial coefficients are given by , satisfy the symmetry , and the Pascal relation . The general term is , and the expansion has terms. For fractional or negative exponents, the generalized binomial series applies: with . The calculator supports both the finite polynomial case and, when requested, the truncated generalized series.
Binomial expansion:
Coefficient: Pascal:
General term: Terms:
Generalized series:
How to Use This Binomial Theorem Calculator
- Enter the base expression in the form (variables or numbers) and specify .
- Choose outputs: full expansion, specific term/degree, or a particular coefficient (e.g., the coefficient of ).
- Click calculate. The tool uses factorial or Pascal recurrences to generate coefficients and simplifies powers.
- Copy the expansion, a requested term, or a series truncation for generalized exponents.
Examples
- Classic: Expand ⇒ coefficients from row 5: giving .
- Mixed signs: ⇒ .
- Chosen coefficient: Coefficient of in is .
- Generalized series (truncated): valid for .
Formula Snippets Ready for Rendering
FAQs
How many terms are in the expansion?
Exactly terms appear in , from to .
What is the coefficient of a specific power?
For , the coefficient of is .
How does Pascal’s triangle relate?
Row lists . Each entry equals the sum of the two above via the Pascal recurrence.
Can the calculator handle fractional exponents?
Yes—use the generalized binomial series and choose a truncation order; it’s valid for .
Why are coefficients symmetric?
Because , matching terms from the beginning and end of the expansion.
What about negative powers?
Use the generalized form with ; the result is an infinite series that can be truncated for approximation.