Polynomial Roots Calculator
Type a polynomial such as x^3 - 6x^2 + 11x - 6 (or enter coefficients) to get every real and complex root up to degree 10, exact fractions and square roots where they exist, multiplicities and the factored form.
The roots of a polynomial are the x values where p(x) = 0.
Use the Polynomial Roots Calculator
Polynomial Input
Use ^ for powers; fractions like 1/2 are fine. Up to degree 10. Or pick a degree and enter coefficients below.
Examples
Type a polynomial or enter coefficients and click Find Roots.
Roots
Factored Form
Rational Root Theorem
Analysis
How polynomial roots are found, by degree
| Degree | Name | Exact method | Roots |
|---|---|---|---|
| 1 | Linear | x = −b / a | 1 real |
| 2 | Quadratic | x = (−b ± √(b² − 4ac)) / 2a | 2 real, or a complex pair when b² − 4ac < 0 |
| 3 | Cubic | Cardano's formula (or a rational root, then a quadratic) | 1 or 3 real |
| 4 | Quartic | Ferrari's method via a resolvent cubic | 0, 2 or 4 real |
| 5+ | Quintic and higher | No general formula (Abel–Ruffini); rational roots, then numerical methods such as Aberth–Ehrlich or Durand–Kerner | n roots counting complex and repeated |
Rational root theorem: any rational root p/q in lowest terms of a polynomial with integer coefficients has p dividing the constant term and q dividing the leading coefficient. Complex roots of real polynomials come in conjugate pairs.
Quick answer
The roots of a polynomial are the x values where p(x) = 0. For x³ − 6x² + 11x − 6, the rational root theorem lists the candidates ±1, ±2, ±3, ±6; x = 1 works, and dividing by (x − 1) leaves x² − 5x + 6 = (x − 2)(x − 3), so the roots are 1, 2 and 3. A quadratic left over is solved with x = (−b ± √(b² − 4ac)) / 2a, and degree 5 and higher usually need a numerical method. A degree-n polynomial has exactly n roots counting complex and repeated ones.
Examples
| Polynomial | Roots | Factored form |
|---|---|---|
| x³ − 6x² + 11x − 6 | 1, 2, 3 | (x − 1)(x − 2)(x − 3) |
| 2x³ − 3x² − 11x + 6 | −2, 1/2, 3 | (x + 2)(2x − 1)(x − 3) |
| x³ − 3x + 2 | −2, 1 (double) | (x + 2)(x − 1)² |
| x² − 2x − 1 | 1 − √2, 1 + √2 | (x² − 2x − 1) |
| x⁴ − 1 | −1, 1, −i, i | (x + 1)(x − 1)(x² + 1) |
Type your polynomial or pick a degree and enter coefficients, then click Find Roots.
Summary
This polynomial roots calculator finds every root (zero) of a polynomial with real coefficients up to degree 10. Type the polynomial the way you write it (2x^3 - 3x^2 - 11x + 6, x³ − 1 or 0.5x^2 + x − 4 = 0) or pick a degree and enter the coefficients. It first applies the rational root theorem and tests each candidate p/q exactly, dividing out every rational root with its multiplicity; a quadratic that remains is solved exactly in square-root form, and anything of degree 3 or more that remains is solved numerically with the Aberth–Ehrlich method (a refinement of Durand–Kerner) and polished with Newton steps. You get each root as an exact value where possible and as a decimal, its multiplicity and whether it is real or complex, the list of rational candidates and the factored form. Everything runs in your browser.
How it works
- Type the polynomial (for example x^4 - 5x^2 + 4) or choose a degree and enter the coefficients from the highest power down to the constant. Fractions like 1/2 are allowed.
- Click Find Roots or press Enter.
- The calculator lists the rational root theorem candidates ±p/q and tests each one exactly, dividing out the roots it finds.
- A remaining quadratic is solved with the quadratic formula in exact square-root form; a remaining cubic or higher is solved numerically (Aberth–Ehrlich).
- Read each root as an exact value and a decimal, with its multiplicity and type, plus the factored form.
Use cases
- Finding the zeros of a cubic or quartic for algebra homework.
- Checking rational root theorem work and synthetic division.
- Solving degree 5 to 10 polynomials that have no general formula.
- Finding complex roots and conjugate pairs.
- Factoring a polynomial from its roots.
- Characteristic polynomials and control-system poles in engineering.