POLYNOMIAL ROOTS GUIDE

Polynomial roots and complex roots

A polynomial root is a value of x that makes f(x) equal zero. Learn how coefficient order, real roots, complex roots, and residual checks work before using a polynomial roots calculator.

FORMULA

f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀; roots solve f(x) = 0

WORKED EXAMPLE

For x² − 5x + 6 = 0, enter coefficients 1, −5, and 6. The roots are 2 and 3 because (x − 2)(x − 3) = 0.

STEP BY STEP

Choose the polynomial degree

The degree is the highest power of x. A quadratic has degree 2, a cubic has degree 3, and higher-degree polynomials continue the same coefficient pattern.

Enter coefficients from highest power to constant

For x⁴ − 5x² + 4, include zero placeholders and enter 1, 0, −5, 0, and 4. Missing middle powers still need a coefficient of zero.

Read real and complex roots

Real roots appear as ordinary numbers and are x-intercepts on a real graph. Complex roots appear in a + bi form and may not cross the x-axis visually.

Check the residual

The residual measures how close f(root) is to zero. A very small residual means the numerical answer is close to satisfying the original polynomial.

Use graphing for a visual check

A graph helps you see real x-intercepts, turning points, and approximate behavior. Use Polynomial Solver when you need coefficient-based root values.