How this calculation works
Multiply the linear polynomial by itself for the chosen nonnegative whole exponent and combine equal powers exactly. This matches the binomial coefficient identity and supports negative or fractional coefficients.
Coefficient of xᵏ = choose(n,k) aᵏ bⁿ⁻ᵏA worked example
(2x − 1)³ = 8x³ − 12x² + 6x − 1. The negative constant alternates the signs of successive terms.
Good to know
Bounded exact rational inputs within ±1,000,000, up to 6 decimal places or integer fraction denominators up to 1,000,000. Large intermediate fractions have a size limit. No arbitrary expression evaluation, multiple variables or general computer-algebra system. Approximate lengths and quadratic square roots are labeled separately. Whole exponents 0–20 only. The zeroth polynomial power is defined as the constant 1, including the zero-polynomial base. No fractional/negative powers or multi-variable input.
Reference: OpenStax: the binomial theorem (opens in a new tab) ↗
Binomial coefficients count how products combine
Expanding (ax+b)ⁿ means choosing either ax or b from each of n factors. To obtain x^k, choose ax from exactly k factors. The number of those choices is the binomial coefficient choose(n,k).
That gives the coefficient choose(n,k)a^k b^(n−k). For (2x−1)³, the coefficients are 8, −12, 6 and −1. The negative constant supplies alternating signs; copying only the positive Pascal-triangle entries would miss them.
The combinations calculator explains the choose(n,k) counting factor. Here that factor is also multiplied by the appropriate powers of a and b, so the expanded coefficients are not generally just a row of Pascal’s triangle.
Rational coefficients and the zeroth power need explicit conventions
Fractional a or b is supported as an exact rational input. A zero a leaves a constant power of b, while a zero b leaves a single power of x when n is positive. Internal zero coefficients remain visible in the output table.
At n = 0 the result is the constant polynomial 1 under the empty-product convention used by this algebra tool, including a zero-polynomial base. It is not presented as a general numerical resolution of every context involving 0⁰.
Polynomial arithmetic can check a small expansion by multiplying the linear factor repeatedly. The expansion calculator supports whole exponents through 20; negative or fractional powers would require a different finite or infinite expansion treatment.
A few questions, answered
Does this support (x+y)ⁿ?
Only a coefficient-defined single-variable form (ax+b)ⁿ is accepted. The constant b cannot be another symbolic variable.
Can n be negative or fractional?
No. Use a whole exponent from 0 to 20.
Why are some coefficients negative?
Signs come from powers of the entered coefficients. A negative b can alternate signs across powers, depending on the exponent.