Binomial Expansion Calculator — Expand (a+bx)n Instantly

Free online binomial expansion calculator with full integer expansions, fractional and negative power series, Pascal's triangle visualization, and step-by-step binomial coefficient breakdown.

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Binomial Expansion Calculator

Enter values for a, b, and the exponent n to expand (a + bx)n using the binomial theorem.

(a + bx)n
Enter a, b, and n then click Calculate Expansion to see the result.

Binomial Expansion Formula Explained

The binomial theorem provides a formula for expanding expressions raised to a power. For (a + b)n, each term involves a binomial coefficient C(n,k), powers of a and b, and follows a precise pattern.

(a + b)n = Σ(k=0 to n) C(n,k) · an−k · bk
C(n,k) = n! / (k! · (n−k)!) — Binomial Coefficient

For General Powers (Negative & Fractional n)

(1 + x)n = 1 + nx + n(n−1)x²/2! + n(n−1)(n−2)x³/3! + …

This infinite series converges for |x| < 1 when n is not a positive integer. The generalized binomial coefficient is C(n,k) = n(n−1)(n−2)…(n−k+1) / k!.

Variable Definitions

  • a — The first term constant in the binomial (a + bx)
  • b — The coefficient of x in the second term
  • n — The exponent (power) to which the binomial is raised
  • C(n,k) — The binomial coefficient "n choose k"
  • k — The term index (0-indexed: k=0 gives the first term)

How to Calculate a Binomial Expansion

Follow these steps to expand any binomial expression using the binomial theorem:

  1. Identify a, b, and n — Write your expression in the form (a + bx)n.
  2. Determine binomial coefficients — For integer n, compute C(n,k) = n!/(k!(n−k)!) for k = 0, 1, …, n. Or use Pascal's triangle row n.
  3. Apply the term formula — Each term is Tk+1 = C(n,k) · an−k · (bx)k.
  4. Simplify each term — Compute the powers and multiply by the coefficient.
  5. Sum all terms — Write the expansion from k=0 to k=n as a sum.

For negative or fractional n, use the generalized binomial series which produces an infinite expansion. This calculator shows the first several terms of the series.

Binomial Expansion Calculator Examples

Example 1: Expand (2 + 3x)4

Using n=4, a=2, b=3. Pascal's triangle row 4: 1, 4, 6, 4, 1.

(2 + 3x)4 = 1·24 + 4·23·(3x) + 6·22·(3x)2 + 4·2·(3x)3 + 1·(3x)4
= 16 + 96x + 216x² + 216x³ + 81x⁴

Example 2: Expand (1 + x)−1

Using the generalized binomial theorem for n = −1, a = 1, b = 1.

(1 + x)−1 = 1 − x + x² − x³ + x⁴ − x⁵ + …
Valid for |x| < 1

Example 3: Find the 3rd term of (x + 2)6

The 3rd term corresponds to k=2: T3 = C(6,2) · x4 · 22 = 15 · x4 · 4 = 60x4.

Example 4: Expand (1 − 2x)1/2 (first 4 terms)

(1 − 2x)1/2 = 1 + (1/2)(−2x) + (1/2)(−1/2)(−2x)²/2! + (1/2)(−1/2)(−3/2)(−2x)³/3! + …
= 1 − x − x²/2 − x³/2 + …

Real-World Binomial Expansion Applications

  • Probability & Genetics: The binomial distribution uses expansions to calculate probabilities of k successes in n trials, essential in genetics for predicting trait inheritance patterns.
  • Calculus Approximations: Binomial series approximate functions like (1+x)1/2 for small x, used in physics for relativistic corrections and perturbation theory.
  • Engineering Series: Taylor and Maclaurin series often reduce to binomial expansions for rational powers, enabling efficient numerical computation.
  • Finance: Compound interest formulas (1+r)n are direct binomial expansions, used to model growth over multiple periods.
  • Physics: Binomial expansions approximate gravitational potentials, electrostatic multipole expansions, and quantum mechanical perturbation series.
  • Computer Algorithms: Binomial coefficients appear in combinatorial algorithms, dynamic programming solutions, and generating function analysis.
  • Statistics: The binomial theorem underpins moment generating functions and the derivation of binomial distribution properties.

People Also Ask

The binomial expansion formula is (a + b)n = Σ(k=0 to n) C(n,k) · an−k · bk, where C(n,k) = n!/(k!(n−k)!) is the binomial coefficient. For non-integer n, the expansion is an infinite series using generalized binomial coefficients: (1 + x)n = 1 + nx + n(n−1)x²/2! + n(n−1)(n−2)x³/3! + ...
Pascal's triangle directly provides the binomial coefficients for expanding (a+b)n when n is a positive integer. Row n (starting with row 0 at the top) contains the coefficients C(n,0), C(n,1), ..., C(n,n). For example, row 5 is 1, 5, 10, 10, 5, 1, which are the coefficients for (a+b)5.
Yes, binomial expansion works for negative powers using the generalized binomial theorem. For (1 + x)−n where n > 0, the expansion is an infinite series: 1 − nx + n(n+1)x²/2! − n(n+1)(n+2)x³/3! + ... The expansion converges for |x| < 1 and is widely used for series approximations.
The binomial coefficient nCr (also written C(n,r) or (n choose r)) equals n!/(r!(n−r)!). It counts the number of ways to choose r items from n distinct items. In binomial expansion, C(n,k) is the coefficient of the an−kbk term. Most scientific calculators have an nCr button for quick computation.
The (k+1)-th term in (a+b)n is Tk+1 = C(n,k) · an−k · bk. For example, to find the 5th term (k=4) of (2+x)8: T5 = C(8,4) · 24 · x4 = 70 · 16 · x4 = 1120x4. Use this formula to extract any term without computing the full expansion.

Frequently Asked Questions

Yes. Switch to "General Power (Series)" mode to expand binomials with fractional exponents like (1+x)1/2 or (a+bx)2/3. The calculator uses the generalized binomial theorem and displays the first several terms of the infinite series expansion.
For negative and fractional powers, the expansion is an infinite series. This calculator displays the first 6 terms (k=0 through k=5) along with an ellipsis (…) to indicate the series continues. This is sufficient for most approximation needs.
nCr (combinations) counts selections where order doesn't matter: C(n,r) = n!/(r!(n−r)!). nPr (permutations) counts arrangements where order matters: P(n,r) = n!/(n−r)!. Binomial expansion uses nCr (combinations) as the binomial coefficients.
This is the geometric series. Using the generalized binomial theorem for n=−1: C(−1,k) = (−1)(−2)...(−k)/k! = (−1)k. So (1+x)−1 = Σ(−1)kxk = 1 − x + x² − x³ + ..., which matches 1/(1+x) for |x|<1.
Absolutely. In genetics, the binomial expansion (p+q)n models genotype frequencies for n offspring, where p and q are allele probabilities. For example, (p+q)2 = p² + 2pq + q² gives the Hardy-Weinberg equilibrium genotype frequencies.
The constant term is the term with no x variable (x0). In (a+bx)n, the constant term is always an (the first term, k=0). For expressions like (x + 1/x)n, the constant term occurs when the powers of x cancel, requiring specific k values.

Binomial Expansion Glossary

Binomial Theorem

The formula for expanding (a+b)n as a sum of terms involving binomial coefficients, powers of a, and powers of b.

Binomial Coefficient

C(n,k) = n!/(k!(n−k)!) — the numerical coefficient of each term in a binomial expansion, also equal to entries in Pascal's triangle.

Pascal's Triangle

A triangular array where each number is the sum of the two above it. Row n provides the coefficients for (a+b)n.

Factorial

n! = n × (n−1) × (n−2) × ... × 1. Used to compute binomial coefficients. 0! = 1 by definition.

Generalized Binomial Theorem

Extends the binomial theorem to any real exponent n, producing an infinite series using generalized binomial coefficients.

Binomial Series

The infinite series expansion of (1+x)n for non-integer n: 1 + nx + n(n−1)x²/2! + ..., converging for |x|<1.

Combinations (nCr)

The number of ways to choose r items from n distinct items without regard to order. Equal to the binomial coefficient C(n,r).

Binomial Distribution

A probability distribution for the number of successes in n independent trials, with probabilities given by terms of (p+q)n where q=1−p.

Editorial Review & Methodology

This binomial expansion calculator was built and reviewed by the NumbrWiz Editorial Team. The binomial theorem is a cornerstone of algebra and combinatorics, verified against standard mathematics curricula including A-Level mathematics, AP Calculus, and college-level discrete mathematics textbooks.

  • Formula verification: Cross-checked against multiple authoritative algebra, combinatorics, and calculus sources including the generalized binomial theorem for non-integer exponents.
  • Edge case testing: Tested with n=0, n=1, negative powers, fractional powers, large integer n values, and zero coefficients.
  • Pascal's triangle validation: Coefficient generation verified against known Pascal's triangle rows and combinatorial identities.
  • Series convergence: For general power mode, the calculator displays terms consistent with the convergence radius |bx/a| < 1.

Transparency note: All calculations run client-side in your browser. No data is ever collected, stored, or transmitted. Results are for educational purposes; verify critical calculations independently. The generalized binomial series is truncated to 6 terms for display purposes.

Page last reviewed: May 2026 · NumbrWiz Editorial Team