Binomial Expansion Calculator step by step · 2026
Expand (a + b)ⁿ with precision — supports integer, fractional (rational) powers, and negative exponents. Get detailed term-by-term expansion, binomial coefficients, Pascal’s triangle insights, and series approximation. Perfect for algebra, calculus, and statistics.
Interactive Binomial Expansion Tool
(2 + 3)^4 = 16 + 96x? waiting for calculation...
How to Use the Binomial Expansion Calculator
Enter numeric values for a and b (real numbers), and an exponent n. The calculator supports:
- Positive integers → finite expansion using binomial theorem & Pascal’s triangle.
- Negative integers & fractions → infinite binomial series (first N terms shown).
- Adjust the "Terms" slider to control series depth for non-integer n.
Comparison of Expansion Methods (2026 Update)
| Method | Best for | Formula | Example (n=4) |
|---|---|---|---|
| Pascal’s Triangle | Small positive integers (n ≤ 10) | Row n+1 coefficients | 1 4 6 4 1 |
| Combination Formula | Any integer n ≥ 0 | C(n,k) aⁿ⁻ᵏ bᵏ | C(4,0)a⁴ + C(4,1)a³b + ... |
| Generalized Binomial Series | Fractional/negative n | (1+x)ⁿ = Σ C(n,k) xᵏ (k≥0) | Works for |b/a| < 1 region |
Step-by-Step Walkthrough: Binomial Theorem Explained
The binomial expansion calculator relies on the fundamental theorem: \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}\) for integer n ≥ 0. For fractional or negative exponents, the expansion becomes an infinite series: \((a+b)^n = a^n \sum_{k=0}^{\infty} \binom{n}{k} (b/a)^k\) where \(\binom{n}{k} = \frac{n(n-1)...(n-k+1)}{k!}\). Our tool shows term-by-term coefficients with full working.
Example 1: Positive integer exponent (n=5)
(2x + 3y)⁵ expands into 32x⁵ + 240x⁴y + 720x³y² + 1080x²y³ + 810xy⁴ + 243y⁵. Our calculator shows each binomial coefficient and power combination.
Example 2: Fractional exponent (sqrt)
√(1+x) = (1+x)^{0.5} = 1 + 0.5x - 0.125x² + 0.0625x³ - ... Up to 5 terms approximation. The tool displays series terms with binomial coefficient for real exponent.
How to Perform Binomial Expansion on TI-84 / Casio fx-991EX
Many students ask “how to do binomial expansion on a calculator?” On TI-84, use the nCr function: for term k, enter nCr(n,k) * a^(n-k) * b^k. Our online tool replicates that logic instantly. For Casio fx-991EX, use the binomial PD function. But with TotalCalcHub’s binomial expansion calculator, you get visual step-by-step without manual keypresses.
Frequently Asked Questions (Binomial Expansion)
Yes! Enter negative exponent (e.g., -3) and the tool will generate the binomial series expansion (infinite series) up to the specified number of terms.
Absolutely — when exponent n is a positive integer, the step-by-step breakdown includes Pascal’s triangle row coefficients.
It uses the generalized binomial theorem. The series converges for |b/a| < 1; more terms yield higher accuracy. Our calculator displays warning if convergence condition is marginal.
Currently a and b accept numeric values. To expand (x+2)³, set a=1, b=2, then multiply variables manually: result: x³ + 6x² + 12x + 8.
Yes, 100% free, no login. Aligned with 2026 AP, A-Level, and college algebra standards.
Binomial expansion specifically deals with (a+b)ⁿ; our tool is specialized for binomials and includes series for fractional/negative exponents.
After expansion, we list each term with index; you can also use the general term formula: Tₖ₊₁ = C(n,k) aⁿ⁻ᵏ bᵏ — our step-by-step shows each term.