Polynomial Root Solver (Real and Complex Roots)
Find all real and complex roots of a polynomial up to degree 20, with exact rational roots, multiplicities and a residual check for each root.
Find every root of a polynomial, real and complex, up to degree 20. Type the polynomial the way you would write it (x^3 - 6x^2 + 11x - 6) or just its coefficients (1, -6, 11, -6). Rational roots such as 1/2 are found exactly with the rational root theorem; the remaining roots are computed numerically and displayed to ten significant digits. Exact rational repeats are grouped; close numerical roots may be less accurate.
How to use this tool
- Enter the polynomial using x as the variable, or list its coefficients from the highest power down (with 0 for missing powers).
- Read the roots: real ones first, then complex conjugate pairs written as a ± bi.
- Use the table to check each root: the residual |P(x)| should be essentially zero.
Formula
P(x) = aₙxⁿ + … + a₁x + a₀ = 0 Rational root theorem: any rational root p/q has p dividing a₀ and q dividing aₙ
The solver tests every candidate p/q exactly, divides out each root it finds, then uses the quadratic formula for a degree-2 remainder or the Durand–Kerner (Weierstrass) iteration for higher degrees, which seeks all complex roots simultaneously; convergence is checked numerically. A polynomial of degree n always has exactly n roots counted with multiplicity.
Example
x³ − 6x² + 11x − 6: the candidates are ±1, ±2, ±3 and ±6. P(1) = 0, and dividing by (x − 1) leaves x² − 5x + 6 = (x − 2)(x − 3), so the roots are 1, 2, 3. x³ − 2 has no rational root; its roots are ∛2 ≈ 1.25992105 and the complex pair −0.6299605249 ± 1.091123636i.
Frequently asked questions
How many roots does a polynomial have?
Exactly as many as its degree, counting complex and repeated roots (the fundamental theorem of algebra). With real coefficients, complex roots come in conjugate pairs, so a cubic always has at least one real root.
What does multiplicity mean?
It is how many times a root repeats. In (x − 1)²(x + 2), x = 1 has multiplicity 2. At an even multiplicity the graph touches the x-axis and turns back instead of crossing it.
Is there a formula for degree 5 and higher?
Not in general. The Abel–Ruffini theorem shows that equations such as x⁵ − x − 1 = 0 cannot be solved with radicals, so numerical methods are the standard approach.
How do I enter missing terms?
In coefficient mode, write 0 for each missing power: x³ − 2 is 1, 0, 0, -2. When you type the expression with x, missing powers are filled in automatically.
Last updated: 2026-10-30