Polynomial Equation Solver

Find all real and complex roots of polynomial equations: exact rational roots first, then numerical roots for the rest.

Your result will appear here

Fill in the fields and press Solve, or pick one of the examples.

About the Polynomial Equation Solver

Finds every root — real and complex, counted with multiplicity — of a polynomial equation of degree up to 64.

Rational roots are found exactly with the rational root theorem and divided out. A remaining quadratic factor is solved exactly with the quadratic formula; anything of higher degree is solved numerically with the Durand–Kerner method and labelled as approximate.

Limitations

There is no general formula in radicals for degree 5 and above (Abel–Ruffini), so such roots are numerical.
Cubic and quartic roots that are irrational are also given numerically rather than in Cardano form.
Coefficients must be rational numbers.

How to use it

Type the polynomial equation, for example x^4 - 1 = 0.
Press Solve.
Exact roots are marked “Exact”; numerical ones “Approximate”.

Example

x^3 - 6x^2 + 11x - 6 = 0

→ x = 1, 2, 3

Formulas used

x = \frac{p}{q},\quad p \mid a_0,\ q \mid a_n

Rational root theorem: any rational root of a polynomial with integer coefficients has this form.

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About the Polynomial Equation Solver

Limitations

How to use it

Formulas used

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About the Polynomial Equation Solver

Limitations

How to use it

Formulas used