(a+b)^n=\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^kVariables
- a, b: binomial terms
- n: nonnegative integer exponent
- k: summation index
- C(n,k): binomial coefficient
How to use this formula
Expands a nonnegative integer power of a binomial.
Important notes
- The coefficients are the entries in row n of Pascal’s triangle.
- The exponents in every term add to n.
Quick example
(a + b)³ = a³ + 3a²b + 3ab² + b³.
Applicability, worked calculation, and verification
Domain and applicability
Apply Binomial Theorem only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.
Assumptions and domain checks
- The coefficients are the entries in row n of Pascal’s triangle.
- For the Binomial Theorem, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.
- For the Binomial Theorem, all variables must lie in the stated domain, and every denominator or inverse operation must be defined.
Do not use this formula when
- Do not use Binomial Theorem when its variable definitions, domain restrictions, or structural assumptions differ from the problem.
Boundary and special cases
- For Binomial Theorem, check zero, negative, and extreme input values before relying on the result.
- When using Binomial Theorem, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values.
Equivalent and alternative forms
- Keep the canonical LaTeX form (a+b)^n=\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k for copying; rearrange only after preserving equivalence and domain restrictions.
Worked example
Expand (2+3)³. The theorem gives 2³+3·2²·3+3·2·3²+3³.
- Set n=3, a=2, and b=3.
- Evaluate the four binomial terms: 8, 36, 54, and 27.
- Add the terms to obtain 8+36+54+27=125.
Independent verification
Direct evaluation gives (2+3)³=5³=125, matching the expansion.
Common mistakes
- Do not omit the index or bounds in Binomial Theorem; changing either one changes which terms are included.
- For the Binomial Theorem, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted.
Continue the workflow
Use Binomial Theorem in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
(a+b)^n=\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-26
Review method: Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Binomial Theorem.
Verified against: OpenStax Algebra and Trigonometry
Automated quality check: Kept noindex until the missing evidence is supplied.
Formula references
- Algebra and Trigonometry 2eOpenStax, Rice University — Reviewed algebra, functions, sequences, trigonometry, and analytic geometry definitions and examples.
Frequently asked questions
What is the Binomial Theorem used for?
Expands a nonnegative integer power of a binomial.
Can I copy this formula as LaTeX?
Yes. Copy (a+b)^n=\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.