How many roots does a polynomial have?

A polynomial of degree n has exactly n roots, counting complex roots and repeated roots according to their multiplicity. A quadratic has 2 roots, a cubic has 3, and a quartic has 4.

What does a complex root look like?

A complex root is shown in the form p + qi, where p is the real part and q is the imaginary part. Complex roots of polynomials with real coefficients always come in conjugate pairs (p + qi and p − qi).

How does this calculator find roots for cubic and quartic equations?

It uses the Durand-Kerner method, an iterative numerical algorithm that refines a set of initial guesses simultaneously until they converge on the true roots — avoiding the need for the complex closed-form Cardano/Ferrari formulas.

Can I solve a simple linear equation with this tool?

Yes, degree 1 is supported as the simplest case: ax + b = 0 solves directly to x = −b/a.

What happens if I enter 0 for the leading coefficient?

A zero leading coefficient means the polynomial is actually a lower degree than selected. The calculator strips leading zero coefficients automatically and solves the resulting lower-degree equation.

Why does a cubic equation sometimes have only one real root shown as complex-looking output?

A cubic always has at least one real root, but the other two can be a complex conjugate pair. When that happens, you'll see one root with a zero imaginary part and two roots with nonzero, opposite-signed imaginary parts.

How accurate are the results?

The Durand-Kerner method iterates until the change between steps is smaller than 10⁻¹², and results are cleaned of residual floating-point noise below 10⁻⁹, giving high-precision results for well-conditioned polynomials.

Can this calculator handle polynomials above degree 4?

No — degree 5 and above have no general closed-form solution (a result known as the Abel–Ruffini theorem), and this calculator is scoped to degree 1–4 to match that boundary and the source problem's typical use cases.

Polynomial Roots Calculator — Find Real and Complex Roots

Our Polynomial Roots Calculator finds every root — real or complex — of any linear, quadratic, cubic, or quartic (degree 1 through 4) polynomial equation. Enter the coefficients of your equation and get all roots instantly, computed using a robust numerical method (Durand-Kerner iteration) that handles every case reliably, including equations whose closed-form solution would otherwise involve unwieldy cubic or quartic formulas.

Quick Answer

Every degree-n polynomial has exactly n roots (real or complex). Enter coefficients for a degree 1–4 equation and this calculator finds all roots at once using a robust numerical method, avoiding the unwieldy closed-form cubic and quartic formulas.

x^2+

= 0

Enter your coefficients, then click Solve.

How to Use the Polynomial Roots Calculator — Solve Equations Up to Degree 4

1
Select the degree of your polynomial (1 to 4).
2
Enter each coefficient, from the highest power of x down to the constant term.
3
Click 'Solve' to find every root of the equation.
4
Review each root — real roots are shown as plain numbers, complex roots are shown in the form p + qi.

Why Use Polynomial Roots Calculator — Solve Equations Up to Degree 4?

Every polynomial of degree n has exactly n roots when complex roots are counted, but finding them by hand gets dramatically harder past degree 2 — cubic and quartic equations have closed-form solutions (Cardano's and Ferrari's formulas), but they're notoriously unwieldy and numerically unstable in edge cases. Our calculator instead uses the Durand-Kerner method, a well-established iterative numerical algorithm that finds all roots of a polynomial simultaneously and converges reliably regardless of whether the roots are real, repeated, or complex — giving you a dependable answer without ever needing to memorize or apply the cubic formula by hand.

Frequently Asked Questions

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