What it does
Type a polynomial, or an equation with polynomials on both sides, and this finds all of its roots (also called zeros): the values of x that make it equal 0. A polynomial of degree n has exactly n roots once you count complex ones and repeats, and all of them are listed, up to degree 6. For one-off equations that aren't polynomials, like 3x = 81, use the equation solver instead.
How the roots are found
- Exact first. Repeated factors are split off exactly, so a double root shows "repeats 2". Every rational root is found and confirmed in exact fractions. A leftover quadratic is solved with the quadratic formula, giving answers like (2 ± √6)/2.
- Closed forms where they're clean. For a leftover cubic the page shows the discriminant and Cardano's formula (one real root) or the sine form from the NIST DLMF (three real roots). Equations like x³ = 2 and quartics with only even powers get their own short forms.
- Numeric roots for the rest. Anything still unsolved is found with the Aberth method, which improves guesses for all the roots at once and converges cubically. Results are shown to 10 significant figures.
- Every root is checked by substituting it back in. Rational roots give exactly 0; decimal roots leave a tiny residual, like 10−15, from rounding.
Rational root theorem: if p/q is a rational root of anxn + … + a0 (integer coefficients, in lowest terms), then p divides a0 and q divides an.
Worked example: 2x³ − 3x² − 11x + 6
The constant term is 6 and the leading coefficient is 2. So p is one of 1, 2, 3, 6 and q is 1 or 2, giving 12 candidates: ±1, ±2, ±3, ±6, ±1/2, ±3/2. Testing them, f(3) = 54 − 27 − 33 + 6 = 0, f(−2) = −16 − 12 + 22 + 6 = 0 and f(1/2) = 1/4 − 3/4 − 11/2 + 6 = 0. The roots are x = −2, 1/2 and 3, and the factored form is (x + 2)(2x − 1)(x − 3).
Useful facts the page uses
| Fact | What it tells you |
|---|---|
| Fundamental theorem of algebra | Degree n means n roots, counting complex roots and repeats |
| Complex conjugate theorem | With real coefficients, complex roots come in pairs a ± bi |
| Descartes' rule of signs | The number of positive roots is at most the number of sign changes, and differs from it by an even number |
| Cubic discriminant Δ | Δ > 0: three different real roots; Δ < 0: one real and two complex |
Limits
- Degree 1 to 6 in a single variable x, with coefficients up to 30 digits.
- From degree 5 up there's generally no formula in square and cube roots, so those roots are decimals unless they're rational or come from a factor of degree 2.
- Roots very close together can lose a few digits of accuracy; the residual column shows how well each one checks out.
Sources
- OpenStax College Algebra 2e, 5.5 Zeros of Polynomial Functions
- NIST Digital Library of Mathematical Functions, §1.11 Zeros of Polynomials
- NIST Digital Library of Mathematical Functions, §4.43 Cubic Equations
- O. Aberth (1973), Iteration methods for finding all zeros of a polynomial simultaneously, Mathematics of Computation 27, 339–344
- Wolfram MathWorld: Abel's Impossibility Theorem