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Binomial Expansion Calculator

Expand (ax + b)^n using the binomial theorem

a =

b =

n =

Binomial Theorem

(p+q)^n = Σ C(n,k)p^(n-k)q^k

The binomial theorem expands powers of a two-term expression without multiplying the binomial repeatedly. Each term contains a binomial coefficient, a decreasing power of the first part, and an increasing power of the second part. The coefficients match Pascal's triangle, while the exponents always add up to n. This structure makes binomial expansion useful for algebra simplification, combinatorics, and pattern recognition.

⚠Note: This calculator limits n to 10 for readable output.

What Is Binomial Expansion?

Binomial expansion rewrites a power of a two-term expression, such as (ax+b)^n, as a polynomial. The binomial theorem gives every coefficient and exponent without multiplying the binomial repeatedly.

Binomial Coefficients

C(n,k) gives the coefficient pattern, the same numbers found in Pascal's triangle.

Power Pattern

The power of ax decreases while the power of b increases from term to term.

Signs

If b is negative, alternating signs can appear depending on the exponent of b.

Polynomial Degree

The expanded polynomial has degree n when a is nonzero.

💡 Example: (x+2)^4 = x^4 + 8x^3 + 24x^2 + 32x + 16.

Applications of Binomial Expansion

Algebra ExpansionPascal TriangleCombinatorics

Frequently Asked Questions

What is a binomial expansion calculator?▼

A binomial expansion calculator expands expressions such as (ax+b)^n into a polynomial using the binomial theorem.

What formula does this tool use?▼

It uses (p+q)^n = Σ C(n,k)p^(n-k)q^k, where C(n,k) is a binomial coefficient.

What is C(n,k)?▼

C(n,k), also called n choose k, is the number of ways to choose k items from n items and gives each term coefficient.

Can this expand negative binomials?▼

Yes. If b is negative, the signs of the expanded terms follow the powers of that negative value.

Why is n limited to 10?▼

The calculator limits n to 10 so the expanded polynomial and step-by-step output stay readable on mobile screens.

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