Polynomial calculator
Add, subtract and multiply polynomials, evaluate with Horner’s method, do synthetic division with the remainder and factor theorems, find rational roots, solve degrees 1–3 exactly and higher degrees numerically, plus a factorised form.
Runs in your browserEvery computation happens in your browser — your data never leaves this device.
Polynomials
Coefficients are written in descending order and separated by spaces or commas; keep zeros for missing powers (x^3 + 1 becomes 1 0 0 1). Everything runs locally.
Result
1 , 2 , 3
3Cardano’s cubic formula1, 2, 3(x - 1)(x - 2)(x - 3)1 , 2 , 3What this tool does
- Work through a polynomial exercise in one pass: add, subtract, multiply, evaluate, do synthetic division and find all roots, then copy the answer into your write-up.
- Check a hand calculation with the Horner intermediates: the evaluation lists every intermediate value, so you can compare them against your own long division step by step.
- Factor quickly: rational roots plus the factor theorem give you forms like (x − 1)(x − 2)(x − 3) without guessing candidates by hand.
- Understand a polynomial before plotting it: degrees above three with no rational root are solved with Durand–Kerner iteration, complex roots included.
Example
Input
p(x) = 2x³ − 3x + 1 (coefficients 2 0 -3 1), x = 2, evaluate
Output
11; Horner intermediates 2 → 4 → 5 → 11; degree 3
Write coefficients in descending order and keep the zeros: 2x³ − 3x + 1 is 2 0 -3 1, and dropping the middle zero changes the polynomial entirely.
Frequently asked questions
How do I enter the coefficients?
In descending order, separated by spaces or commas, with 0 for missing powers: x³ + 1 becomes 1 0 0 1 and 2x³ − 3x + 1 becomes 2 0 -3 1. Brackets and line breaks are ignored, so pasting an array from code works too.
Which root-finding methods are used?
Linear equations use −b/a, quadratics use the quadratic formula and cubics use Cardano’s formula (with the trigonometric branch when the discriminant is negative, which keeps all three real roots stable). Degree four and above are first deflated by rational roots, then solved with the same formulas if at most cubic remains, and finally with Durand–Kerner iteration.
How are rational roots found?
By the rational root theorem: if a reduced fraction p/q is a root of an integer-coefficient polynomial, then p divides the constant term and q divides the leading coefficient. The tool enumerates exactly those candidates and verifies each one, so nothing is missed or invented; non-integer coefficients skip this step.
How accurate are synthetic division and the high-degree results?
Synthetic division is exact integer/rational arithmetic. High-degree polynomials without rational roots go through numeric iteration that converges to about 8 significant digits, usually within 1e-8. If you need a strict radical form, keep the degree at three or below.
Is there a degree limit?
Up to degree 20. Beyond that the conditioning of the iteration degrades quickly and the printed roots stop being meaningful, so the tool reports that instead of showing unreliable digits.
Keywords：polynomialpolynomial calculator多项式运算多项式求根hornersynthetic division综合除法rational root有理根factor theorem因式定理factorization因式分解cubic formula