How to use it
- Choose How to enter the polynomial: as a polynomial, or as a list of coefficients.
- Type the polynomial in Polynomial in x, or the numbers from the highest power down in Coefficients, from the highest power down.
- Read Summary and the The roots table. Check tells you whether every root passes the test by substitution.
- Go through How it is worked out to see the rational root test, each division and the method for the rest.
Rational roots, closed forms and numeric roots
A rational root p/q of a polynomial with whole coefficients has p dividing the constant term and q dividing the leading coefficient. Each candidate is tested by synthetic division, and each root found is divided out, so the polynomial gets shorter. A quadratic that is left is solved with the quadratic formula, and its square roots are written as k√m. A polynomial in x² alone is solved through y = x² when its y-roots are rational.
x = (−b ± √D) ÷ 2a, with D = b² − 4ac
p(x) = (x − r)·q(x) + remainder, with remainder 0 for a root r- Degree 1 and 2 are always solved exactly; a cubic or a quartic is solved exactly only when the rational root test or the x² form reduces it to a quadratic.
- Other polynomials, such as x³ − 2 or a quintic, are solved numerically: Durand–Kerner finds all roots at once, and Newton steps refine them.
- Each numeric root is checked by substituting it into the original polynomial; a relative residual near 10⁻¹⁵ means it is accurate.
What the calculator does not cover
The calculator handles one polynomial in x of degree 1 to 6 with real coefficients. Multiple roots that are not rational are found numerically, so their decimal values may be less accurate than the others.
- Polynomials of degree 7 or more, and polynomials in several variables, are refused.
- Cardano's formula for cubics and Ferrari's formula for quartics are not used; the numeric method covers those cases.
- Divisors of very large constant terms are only partly searched, and the page says so when that happens.
Worked example: 2x³ − 3x + 1 = 0
The candidates are ±1 and ±1/2. Synthetic division by x − 1 gives the coefficients 2, 2, −1 with remainder 0, so 1 is a root and the quotient is 2x² + 2x − 1.
For the quotient, D = 2² − 4 × 2 × (−1) = 12 = 2²·3, so its roots are (−1 ± √3)/2, about 0.366 and −1.366. All three roots are real.
| Step | Result |
|---|---|
| Rational root | 1 |
| Quotient after dividing by x − 1 | 2x² + 2x − 1 |
| Roots of the quotient | (−1 ± √3)/2 ≈ 0.366 and −1.366 |
| Summary | Degree 3: 3 real roots |
Frequently asked questions
What is the rational root test?
It lists the only rational numbers that can be roots of a polynomial with whole coefficients: p/q, where p divides the constant term and q divides the leading coefficient. The calculator tests each one by synthetic division.
Why are some roots shown as decimals?
A root that is not rational and has no simple closed form, such as the cube root of 2, is found numerically and shown as a decimal. The residual column shows how close it is to a true root.
What does the residual mean?
The residual is |p(x)| at the root, divided by the sizes of the terms of the polynomial. A value near 10⁻¹⁵ means the root is correct to the limit of the arithmetic; a larger value means the root is less accurate.
How do I enter a polynomial with fractions?
Use the coefficient list and write each fraction as a number, such as 1/2 for one half. In the typed form, write the fraction in front of the power without brackets: 1/2x^2 + 1. Products of brackets are not expanded, so write the polynomial out.
Are my numbers sent to a server?
No. The roots are computed in your browser, and nothing you type is sent to us.
Sources
- Wikipedia: Rational root theorem
- Wikipedia: Synthetic division
- Wikipedia: Durand–Kerner method
- Wikipedia: Cubic equation
Figures and text last checked on Tuesday, November 10, 2026.