How to use the binomial theorem expansion calculator
This calculator expands a binomial of the form (ax + b)ⁿ. Enter a, b and the exponent n to get the full expansion plus a table of binomial coefficients and term coefficients, and you can also look up a single term on its own.
The rule used is the binomial theorem itself: the k-th term is C(n, k) × aⁿ⁻ᵏ × bᵏ × xⁿ⁻ᵏ. Binomial coefficients are built by sequential multiplication from the previous term instead of computing factorials directly, which keeps intermediate values small and the coefficients exact even for larger n.
A checking line is included as well. Substituting x = 1 means every coefficient added together must equal (a + b)ⁿ, and because the calculator prints that value you can add the table column and compare at once. When b is 0 only one term survives, so the count of non-zero terms is shown next to the total number of terms.
The exponent runs from 0 to 30 and a and b are accepted up to an absolute value of 1,000. Coefficients grow very quickly once binomial coefficients and powers combine, so if one passes the limit of exactly representable integers the calculation stops and asks for smaller values. No thousands separators are used inside the expansion or the term column.
Frequently Asked Questions
The k-th term of (ax + b)ⁿ is C(n, k) × aⁿ⁻ᵏ × bᵏ × xⁿ⁻ᵏ. For example (2x − 3)³ expands to 8x³ − 36x² + 54x − 27.
Substituting x = 1 makes the sum of all coefficients equal (a + b)ⁿ. The calculator prints that value, so adding up the coefficients in the table is an instant check.
Binomial coefficients multiplied by powers grow very fast. The limit of 30 keeps the coefficients inside the range that can be represented exactly, and if a coefficient still passes that limit a message is shown.