Math calculators

Polynomial calculator

Find all roots of a polynomial, evaluate it, add, subtract, multiply or divide polynomials, and take derivatives and integrals.

Inputs

p coefficients, highest power firstp(x) = x^3 - 7x + 6

Results

Methodology

Conventions

Coefficients are listed from the highest power down: 1, 0, −2, 1 is x³ − 2x + 1.
All roots are found together with the Aberth–Ehrlich iteration and polished; values below 10⁻¹⁰ are shown as zero.
Division is polynomial long division; integration sets the constant of integration to zero.

References

Aberth, Iteration methods for finding all zeros of a polynomial simultaneously (1973)

Formulas

Evaluation (Horner): p(x) = (\cdots(a_n x + a_{n-1})x + \cdots)x + a_0
Division: p = q \cdot d + r,\quad \deg r < \deg d
Aberth step: z_k \leftarrow z_k - \frac{w_k}{1 - w_k \sum_{j \ne k} \frac{1}{z_k - z_j}},\quad w_k = \frac{p(z_k)}{p'(z_k)}