$Polynomial Equations$

Polynomial Equations

Quartic equations and polynomials up to degree 10: real and complex roots with multiplicity.

For example, 1 −5 5 5 −6 means x⁴ − 5x³ + 5x² + 5x − 6 = 0. Use zero for a missing power. Fractions as 1/2.

This calculator solves polynomial equations up to degree 10, including quartic equations. It finds all real and complex roots and shows their multiplicity.

How it works

It finds rational roots exactly by trying fractions p/q, where p divides the constant term and q divides the leading coefficient.
It divides the polynomial by the factors found.
If a quadratic remains, it solves it with the discriminant. Other roots are found numerically and marked with ≈.

Example. x⁴ − 5x³ + 5x² + 5x − 6 = 0 has the roots −1, 1, 2 and 3.

An equation of degree n has exactly n complex roots counted with multiplicity. For degree 5 and above there is no general formula in radicals.