Does this calculator find complex roots?

Yes. If a polynomial has non-real roots, they are shown in a + bi form. Complex roots are common when the real graph does not cross the x-axis for every root.

What does the residual mean?

The residual is the size of f(root). A very small residual means the calculated root is close to making the polynomial equal zero.

Can I solve quartic and higher-degree equations?

Yes. The workspace supports degree 1 through 10. For degree 4 and above it uses numerical root-finding instead of long symbolic formulas.

When should I use the Graphing Calculator instead?

Use Graphing Calculator when you want to see the curve, x-intercepts, turning behavior, and approximate visual roots. Use Polynomial Solver when you have coefficients and need root values.

POLYNOMIAL ROOT CALCULATOR

Polynomial solver for real and complex roots.

Q: How do I enter a polynomial?

Choose the degree, then enter coefficients from the highest power down to the constant term. For x² − 5x + 6, enter 1, −5, and 6.

Enter coefficients for degree 1 to 10 polynomials and find real or complex roots numerically. Use it for quadratic, cubic, quartic, and higher-degree equations when you need a coefficient-based roots calculator.

COEFFICIENT WORKSPACE

Build your polynomial

Degree 4

Polynomial degree

YOUR POLYNOMIALx⁴ − 5x² + 4 = 0

a⁴x⁴a³x³a²x²a¹x¹a₀constant

Try a sample:

Polynomial roots from coefficients

Choose the degree, enter coefficients from the highest power down to the constant term, and solve f(x) = 0.

Real and complex roots included

Real roots are shown as ordinary numbers. Non-real roots are displayed in a + bi format when the graph does not cross the x-axis.

Residual check for accuracy

The residual shows how close f(root) is to zero, which is useful for repeated roots and higher-degree numerical answers.

POLYNOMIAL ROOTS EXPLAINED

How to use this polynomial roots calculator.

A polynomial root is a value of x that makes f(x) equal zero. This workspace is built for coefficient form, so it is best when you already know the coefficients of xⁿ, xⁿ⁻¹, and the constant term.

COEFFICIENT ORDER

Enter highest power first

For x² − 5x + 6, enter 1, −5, and 6. For x⁴ − 5x² + 4, enter 1, 0, −5, 0, and 4.

REAL ROOTS

Where the graph crosses

Real roots are x-intercepts of the polynomial graph. Use the Graphing Calculator when you want to see the curve visually.

COMPLEX ROOTS

When roots contain i

Some polynomials have roots like 2 + 3i. They are valid roots even though they do not appear as x-axis crossings on a real graph.

QUADRATIC EXAMPLE

x² − 5x + 6 = 0

Use degree 2 with coefficients 1, −5, 6. The roots are 2 and 3 because (x − 2)(x − 3) = 0.

QUARTIC EXAMPLE

x⁴ − 5x² + 4 = 0

Use degree 4 with coefficients 1, 0, −5, 0, 4. The sample gives four roots: −2, −1, 1, and 2.

RELATED WORKFLOWS

Use the right SolveGrid tool for the next step.

Polynomial roots are one part of algebra work. Use graphing for visual checks and equation solving for lower-degree exact workflows.

Polynomial roots guide Graphing Calculator Equation Solver

Polynomial Solver

Enter polynomial coefficients from the highest power to the constant term, then calculate real and complex roots numerically with residual checks for degree 1 through 10 equations.

Polynomial roots from coefficients

Quadratic, cubic, quartic, and higher-degree equations

Real and complex numerical roots with residual checks

KEY FUNCTIONS

Degree 1 through 10 polynomial inputCoefficient fields ordered from highest power to constant termReal roots and complex roots in a + bi formatResidual value for each calculated rootQuick samples for quadratic, cubic, quartic, and degree 5 polynomialsHelpful explanations for coefficient order, real roots, and complex roots

Browse math & algebra tools →

Polynomial solver for real and complex roots.

Build your polynomial

Polynomial roots from coefficients

Real and complex roots included

Residual check for accuracy

How to use this polynomial roots calculator.

Enter highest power first

Where the graph crosses

When roots contain i

x² − 5x + 6 = 0

x⁴ − 5x² + 4 = 0

Use the right SolveGrid tool for the next step.