What degrees does this calculator support?

It supports polynomials of degree 1 (linear), 2 (quadratic), 3 (cubic), and 4 (quartic). Degree 0 (constant) is not meaningful for root-finding.

What happens if there are complex roots?

Complex roots are displayed in standard a + bi form (e.g., 1.5 + 2.3i). Every polynomial of degree n has exactly n roots when counted over the complex numbers, including multiplicities.

Why must the leading coefficient be non-zero?

If the leading coefficient is 0, the polynomial degenerates to a lower degree. The calculator will warn you and ask you to either increase the degree or enter a non-zero leading coefficient.

How accurate are the results?

For degrees 1 and 2 the results are algebraically exact (within floating-point precision). Cubic and quartic roots use the classical closed-form formulas (Cardano's method and Ferrari's method), which can accumulate small floating-point errors for extreme coefficient values.

What is the discriminant?

The discriminant is a value derived from the coefficients that tells you the nature of the roots without computing them. Positive discriminant → all real distinct roots; zero → repeated roots; negative → some complex roots (for degree 2 and 3).

Can I find roots of polynomials of degree 5 or higher?

No. By Abel's impossibility theorem, there is no general closed-form algebraic formula for degree 5 and above. Numerical methods would be needed, which are beyond the scope of this tool.

Polynomial Roots Calculator

Name: Polynomial Roots Calculator
Availability: InStock
Author: Nham Vu

Enter coefficients of a polynomial up to degree 4 and instantly find all real and complex roots.

Polynomial Input

Degree

Equation p(x) = …

Enter coefficients and click Find Roots to see results.

Summary

Enter coefficients of a polynomial up to degree 4 and instantly find all real and complex roots.

How it works

Select the degree of your polynomial (1 through 4).
Enter the coefficients for each term, from the highest-degree term down to the constant.
Click "Find Roots" or press Enter to compute.
All roots appear immediately, labeled as real or complex (a + bi form).
Use the discriminant and root summary to understand the nature of the solutions.

Use cases

Solve quadratic equations without manually applying the quadratic formula.
Find all three roots of a cubic equation for algebra or engineering problems.
Verify hand-calculated polynomial roots quickly.
Explore how changing coefficients shifts the roots of a polynomial.
Identify whether a polynomial has real or complex roots before graphing.
Check factored form by confirming the roots match expected values.
Solve linear (degree 1) equations in a unified interface.
Find quartic roots for advanced mathematics or physics problems.

Frequently Asked Questions

Last updated: 2026-06-10 · Reviewed by Nham Vu