Does the calculator show exact roots?

Rational roots found exactly are displayed as reduced fractions. Quadratic irrational and higher-degree non-rational roots are displayed as numerical approximations.

Why do some results say approximately?

Most polynomials of degree five or higher do not have a general solution by radicals. A numerical root method is therefore used for factors that cannot be reduced by exact rational roots or a final linear or quadratic factor.

How do I enter a missing polynomial term?

Enter 0 for its coefficient. For x cubed minus 4x plus 1, enter coefficients 1, 0, negative 4, and 1.

Are fractions and decimals accepted?

Yes. Each coefficient may be an integer, a finite decimal, or a fraction such as negative 3 over 4.

Polynomial Root Calculator — Real and Complex Zeros with Steps

Enter coefficients for a polynomial of degree 1–10. The solver finds every real and complex zero locally in your browser; your coefficients are not sent anywhere.

Get all roots counted with multiplicity, exact rational values when detected, method steps, real/complex counts, and a numerical substitution check.

Enter polynomial coefficients

Roots and verification

All zeros, counted with multiplicity

3 roots found

Polynomialx³ − 6x² + 11x − 6 = 0

Root types3 real; 0 non-real complex

Degree check3 roots for degree 3, including multiplicity

Residual check—

Step-by-step solution

How polynomial roots are found

A zero or root is a value r for which P(r)=0. The corresponding factor is (x−r). This calculator combines exact algebra with numerical approximation so it can cover linear through degree-10 polynomials.

1. Normalize the input

Coefficients are read in descending powers. Common factors do not change the zeros, so the equation may be scaled before solving.

2. Find exact rational zeros

When coefficients can be represented as manageable integers, the Rational Root Theorem supplies candidates ±p/q. Exact hits are removed by synthetic division.

3. Solve small remaining factors

A remaining linear factor is solved directly. A quadratic factor uses x = (−b ± √(b²−4ac))/(2a), including complex results.

4. Approximate higher-degree roots

An Aberth-style simultaneous complex iteration finds any remaining roots. Newton polishing improves each result, and substitution measures the scaled residual.

Root count and conjugate pairs

P(x) = a_n(x − r₁)(x − r₂) ··· (x − r_n)

The Fundamental Theorem of Algebra guarantees n complex roots for degree n, counted with multiplicity. If all coefficients are real, every non-real root a+bi is accompanied by its conjugate a−bi.

Worked examples

Three distinct real zeros

For x³−6x²+11x−6, candidate 1 gives zero. Synthetic division leaves x²−5x+6, which factors as (x−2)(x−3). The roots are 1, 2, and 3.

Real and complex zeros

x³+1=(x+1)(x²−x+1). The real root is −1; the quadratic gives 0.5 ± 0.8660254i.

A repeated zero

x⁴−4x³+6x²−4x+1=(x−1)⁴. The value 1 is one distinct zero with multiplicity 4.

No real zeros

The roots of x⁴+1 are approximately ±0.7071068 ± 0.7071068i. All four are non-real and occur in conjugate pairs.

Calculation method, privacy, and limits

All work runs in this page. Coefficients are never uploaded or stored. Rational-root tests use exact integer arithmetic when the entered fractions can be normalized safely; other roots are numerical approximations shown to up to 10 significant digits.

The calculator accepts degree 1–10, limits each coefficient to 14 digits and an absolute value of 10¹², and rejects coefficient scaling that would make the monic polynomial numerically unsafe. The residual is scaled against the coefficient magnitudes, but a small residual does not guarantee many accurate digits for a severely ill-conditioned polynomial or tightly clustered roots. Last reviewed: 4 August 2026.

Polynomial root calculator FAQs

How many roots does a polynomial have?

A degree-n polynomial has exactly n complex roots when multiplicity is counted. A repeated value occupies more than one of those n root positions.

Can a polynomial have complex roots?

Yes. With real coefficients, non-real roots occur in conjugate pairs: if a+bi is a root, then a−bi is also a root.

What is a repeated root?

A root has multiplicity m when its factor appears m times. For example, (x−2)³ has root 2 with multiplicity 3.

Does the calculator return exact answers?

Exact rational roots are kept as reduced fractions when detected. Roots from a final quadratic or a higher-degree numerical solve are decimal approximations, including complex values.

Why is a numerical method needed?

There is no general solution by radicals for arbitrary degree-five and higher polynomials. Numerical root finding also gives a practical common method for irreducible cubic, quartic, and higher-degree factors.

How do I enter a missing term?

Enter zero for its coefficient. For x³−4x+1, enter 1, 0, −4, 1 from the highest power to the constant.

What does the residual check mean?

The calculator substitutes each displayed numerical root into the original polynomial. It scales the remaining error by the sizes of the coefficients and powers of the root; smaller is better.