Binomial Expansion Calculator
Expand a powered binomial with exact integer arithmetic, preserve signs and variable exponents, inspect Pascal coefficients and isolate any selected term.
Binomial Expansion Calculator inputs
Every material assumption is visible and editable. Invalid or internally inconsistent entries are stopped before a result is shown.
Binomial Expansion Calculator results
How the binomial expansion calculator works
The binomial expansion calculator is designed for one specific search intent. Its formulas, controls, validation, outputs and explanation are tailored to that task rather than borrowed from another calculator.
The binomial theorem
The binomial expansion calculator applies the theorem term by term. For index k, the coefficient is C(n,k), the first expression is raised to n−k, and the second is raised to k. Numeric coefficients and variable powers are combined exactly. This is more reliable than floating-point factorial formulas when coefficients become large.
Exact coefficient engine
All combinatorial and coefficient calculations use integer arithmetic capable of retaining values larger than JavaScript’s ordinary safe-integer range. The exponent is limited to 60 to keep the displayed polynomial practical while still covering substantial algebra examples. Negative second coefficients automatically alternate signs where mathematically required.
Selected term and Pascal row
Choose k from zero through n to inspect one term without manually expanding the whole expression. The binomial expansion calculator reports the selected coefficient and variable powers, and also lists the Pascal-triangle row. Remember that a textbook “third term” corresponds to k = 2 because indexing begins at zero.
Using variable powers
The inputs can represent forms such as (2x² − 3y)⁵ rather than only (x+y)ⁿ. The engine multiplies the base variable powers by each term exponent. It is intended for two algebraic monomials; it does not symbolically simplify nested functions or expressions containing more than two additive terms.
Reading the expanded result
For (2x − 3y)⁵, each term combines a Pascal coefficient with a power of 2, a power of −3, a descending power of x and an ascending power of y. The selected-term panel is useful when an exercise asks for one coefficient rather than the entire expansion. Check whether the question numbers the first term as term one while the calculator indexes it with k equal to zero.
Common algebra mistakes
A negative sign belongs to the second coefficient and is raised to k, so its sign changes with term parity. The combination coefficient is not the final numeric coefficient until powers of both numeric bases are included. Variable exponents also multiply: if the first monomial contains x², its power in a term is 2(n−k). Copy the complete expanded result carefully because very large coefficients can make transcription errors easy.
Binomial Expansion Calculator questions
Are the coefficients rounded?
No. Integer coefficients are calculated exactly.
What does k mean?
It is the zero-based term index in the binomial theorem.
Can the second coefficient be negative?
Yes, and the expanded signs are handled automatically.
Can this expand three terms?
No. This calculator is deliberately limited to a binomial, which has exactly two terms.