Binomial Expansion Calculator
Verified Calculation Engine
The online Binomial Expansion Calculator helps you calculate instantly and solve problems related to Binomial Theorem. This tool provides accurate results using standard formulas and step-by-step calculation; you can view the formula with example in the calculator where available. Whether you are a student, teacher, or professional, this calculator simplifies complex calculations and saves time. Enter the required values below and get instant results. Results are shown clearly, with optional step-by-step explanation where applicable. The tool is free to use and works in any modern browser—no download or installation required. Bookmark this page for quick access whenever you need reliable Math Numbers calculations.
Expands (a + b)^n using binomial theorem: (a + b)^n = Σ(k=0 to n) nCk a^(n-k) b^k.
(a + b)^n = Σ(k=0 to n) nCk a^(n-k) b^k
= nC0 a^n + nC1 a^(n-1) b + nC2 a^(n-2) b² + ... + nCn b^n
Binomial theorem expands (a+b)^n as sum of terms with binomial coefficients
Inputs
Results
Worked Examples
Example 1: Basic Expansion
Expand (x + y)^3
- a: 1
- b: 1
- n: 3
- expansion: x³ + 3x²y + 3xy² + y³
- terms: x³, 3x²y, 3xy², y³
- numberOfTerms: 4
Example 2: Board Level
Expand (2x - 3y)^4
- a: 2
- b: -3
- n: 4
- expansion: 16x⁴ - 96x³y + 216x²y² - 216xy³ + 81y⁴
- terms: 16x⁴, -96x³y, 216x²y², -216xy³, 81y⁴
- numberOfTerms: 5
About this calculator
Overview
Binomial Expansion Calculator evaluates Binomial Expansion for the math / binomial-theorem library. Primary input cue: First Term a [required]: First term. Method anchor: (a + b)^n = Σ(k=0 to n) nCk a^(n-k) b^k = nC0 a^n + nC1 a^(n-1) b + nC2 a^(n-2) b² + ... + nCn b^n. Binomial theorem expands (a+b)^n as sum of terms with binomi.
When to use
Appropriate when binomial expansion is the target quantity and the form fields match your known data.
Inputs explained
- First Term a [required]: First term
- Second Term b [required]: Second term
- Power n [required]: Power to raise binomial to
Formula / method
(a + b)^n = Σ(k=0 to n) nCk a^(n-k) b^k = nC0 a^n + nC1 a^(n-1) b + nC2 a^(n-2) b² + ... + nCn b^n. Binomial theorem expands (a+b)^n as sum of terms with binomial coefficients (calculator id: binomial-expansion-calculator).
Worked example
Worked values: a = 1, b = 1, n = 3 → expansion = x³ + 3x²y + 3xy² + y³, terms = ['x³', '3x²y', '3xy²', 'y³'], numberOfTerms = 4
Interpreting results
If binomial expansion looks implausible, re-check units and which field is optional vs required.
Assumptions
- The wired execution function for `binomial-expansion-calculator` is the source of numerical behavior.
- Inputs for `binomial-expansion-calculator` follow the on-page labels; values/shapes must match this operation.
- Real-number arithmetic is used unless binomial expansion explicitly needs integers.
Limitations
- Display rounding can differ slightly from a CAS/symbolic exact form of binomial expansion.
- Edge cases outside the intended domain of binomial expansion may error or look unstable—treat them as out of scope.
- Complex/branch cuts and undefined points are not always interactively annotated for binomial expansion.
Important note
Educational mathematics tool. Check edge cases and domain restrictions when results look unexpected.
How to Use This Calculator
- Enter the required values in the input fields.
- Click the Calculate button.
- View the computed result instantly.
Formula Used
(a + b)^n = Σ(k=0 to n) nCk a^(n-k) b^k = nC0 a^n + nC1 a^(n-1) b + nC2 a^(n-2) b² + ... + nCn b^n. Binomial theorem expands (a+b)^n as sum of terms with binomial coefficients (calculator id: binomial-expansion-calculator).
Example Calculation
Worked values: a = 1, b = 1, n = 3 → expansion = x³ + 3x²y + 3xy² + y³, terms = ['x³', '3x²y', '3xy²', 'y³'], numberOfTerms = 4
Frequently Asked Questions
What is Binomial Expansion Calculator?
- Binomial Expansion Calculator evaluates Binomial Expansion for the math / binomial-theorem library. Primary input cue: First Term a [required]: First term. Method anchor: (a + b)^n = Σ(k=0 to n) nCk a^(n-k) b^k = nC0 a^n + nC1 a^(n-1) b + nC2 a^(n-2) b² + ... + nCn b^n. Binomial theorem expands (a+b)^n as sum of terms with binomi.
How does Binomial Expansion Calculator work?
- (a + b)^n = Σ(k=0 to n) nCk a^(n-k) b^k = nC0 a^n + nC1 a^(n-1) b + nC2 a^(n-2) b² + ... + nCn b^n. Binomial theorem expands (a+b)^n as sum of terms with binomial coefficients (calculator id: binomial-expansion-calculator).
Why use this Math Numbers calculator?
- Appropriate when binomial expansion is the target quantity and the form fields match your known data.