Type any polynomial up to degree 10, as an expression like 2x^3 − 5x + 1 or as a list of coefficients, and get every root: real and complex, with exact fractions where they exist, multiplicities, the factored form and a graph.
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Examples: 2x^3 - 3x + 1 · (x-1)^2(x+4) · x^5 = 32 · or coefficients from highest power: 1, 0, -2, 1. Degree up to 10.
Enter your values to see the result.
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How to use the Polynomial Root Finder
Enter the polynomial, for example x^4 + 3x^3 + x^2 − 7x − 30, or its coefficients from the highest power down.
Choose the number of significant digits for the results.
Press Find roots (results also update as you type).
Read the roots and the factored form, including complex roots written as a ± bi.
Check the graph and table for multiplicities and residuals.
The Durand–Kerner method
For a polynomial p(x) of degree n with leading coefficient aₙ, start with n distinct guesses z₁…zₙ (points on a circle in the complex plane) and update all of them at once:
zk ← zk − p(zk) ÷ ( aₙ · ∏j≠k (zk − zj) )
Each round uses the latest estimates of the other roots; for simple roots convergence is quadratic. Because all roots are found together, the method naturally returns complex roots too. Afterwards, Newton's method z ← z − p(z)/p′(z) polishes each root, and exact rational roots are confirmed with the rational root theorem (any rational root p/q in lowest terms has p dividing the constant term and q dividing the leading coefficient).
Root facts by degree
Degree
Name
Roots
Real roots possible
Closed-form formula?
1
Linear
1
1
Yes: x = −b/a
2
Quadratic
2
0 or 2
Yes (quadratic formula)
3
Cubic
3
1 or 3
Yes (Cardano)
4
Quartic
4
0, 2 or 4
Yes (Ferrari)
5+
Quintic and higher
n
n, n−2, n−4, …
No general formula (Abel–Ruffini)
Vieta's formulas give a quick check: the sum of all roots equals −aₙ₋₁/aₙ and their product equals (−1)ⁿ·a₀/aₙ.
Frequently asked questions
How many roots does a polynomial have?
A polynomial of degree n has exactly n roots in the complex numbers when roots are counted with multiplicity (the fundamental theorem of algebra). Some may be real and some complex; complex roots of a polynomial with real coefficients always come in conjugate pairs a ± bi.
How does this root finder work?
Rational roots are first found and confirmed with exact fraction arithmetic and divided out. The remaining roots are computed with the Durand–Kerner (Weierstrass) method, which refines all roots at once using z ← z − p(z) ÷ (aₙ ∏(z − other roots)), and each is polished with Newton's method.
Is there a formula for the roots of degree 5 or higher?
No. The Abel–Ruffini theorem shows there is no general formula using radicals for degree 5 and above, so roots are found numerically. Quadratics, cubics and quartics have formulas, but numerical methods are usually more practical.
What is the multiplicity of a root?
It is how many times the factor (x − r) appears. For p(x) = (x − 1)³(x + 2)², the root 1 has multiplicity 3 and −2 has multiplicity 2. At a root of even multiplicity the graph touches the x-axis without crossing it.
How accurate are the roots?
Rational roots are exact. Other roots are accurate to roughly 12 to 15 significant digits for simple roots; repeated irrational roots are less precise because they are numerically sensitive. The table shows |p(root)| so you can judge the accuracy.
What input formats are accepted?
Expressions with ^ or ² for powers and implied multiplication, such as 3x^2 − x/2 + 1, products like (x − 2)(x² + 1), equations like x^3 = 8, or a list of coefficients from the highest power, such as 1, 0, −2, 1 for x³ − 2x + 1.
What is the factored form over the reals?
It writes the polynomial as its leading coefficient times linear factors (x − r) for the real roots and quadratic factors x² − 2ax + (a² + b²) for each complex pair a ± bi, so all coefficients stay real.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.