Polynomial Equation Root Calculator

A polynomial equation root calculator helps you find all the roots (solutions) of a polynomial equation. Polynomials are mathematical expressions with variables raised to whole number powers and coefficients. Roots are the values of the variable that make the polynomial equal to zero.

What is a Polynomial Root?

A polynomial root is a value of the variable that makes the polynomial equation equal to zero. For example, in the equation x² - 5x + 6 = 0, the roots are x = 2 and x = 3 because these values satisfy the equation.

Polynomials can have different numbers of roots depending on their degree. A linear polynomial (degree 1) has one root, a quadratic polynomial (degree 2) has two roots, and so on. Higher-degree polynomials can have complex roots.

How to Find Polynomial Roots

There are several methods to find the roots of a polynomial equation:

Factoring: Express the polynomial as a product of simpler polynomials and solve for the roots.
Quadratic Formula: For quadratic equations (degree 2), use the formula x = [-b ± √(b² - 4ac)] / (2a).
Synthetic Division: A method for dividing a polynomial by a linear factor to find roots.
Numerical Methods: Approximate roots using iterative methods like Newton's method.
Graphical Methods: Plot the polynomial and find where it crosses the x-axis.

Our calculator uses a combination of these methods to find all roots of the polynomial equation you provide.

Using the Calculator

To use the polynomial equation root calculator:

Enter the coefficients of your polynomial in the input fields. For example, for the polynomial 2x³ - 5x² + 3x - 7, enter 2 for x³, -5 for x², 3 for x, and -7 for the constant term.
Select the degree of your polynomial from the dropdown menu.
Click the "Calculate Roots" button to find all roots of the polynomial.
The calculator will display all real and complex roots of the polynomial.

The calculator provides a clear explanation of the results and shows the polynomial equation with the roots highlighted.

Examples

Example 1: Quadratic Polynomial

Find the roots of the polynomial x² - 5x + 6 = 0.

Using the quadratic formula: x = [5 ± √(25 - 24)] / 2 = [5 ± 1] / 2.

Roots: x = 3 and x = 2.

Example 2: Cubic Polynomial

Find the roots of the polynomial x³ - 6x² + 11x - 6 = 0.

Using factoring: (x - 1)(x - 2)(x - 3) = 0.

Roots: x = 1, x = 2, and x = 3.

FAQ

What is the difference between a root and a solution?

In the context of polynomial equations, "root" and "solution" mean the same thing—the values of the variable that satisfy the equation.

Can a polynomial have complex roots?

Yes, polynomials can have complex roots when the discriminant is negative (for quadratic equations) or when the polynomial cannot be factored into real roots.

How many roots can a polynomial have?

A polynomial of degree n can have up to n roots, counting multiplicities. For example, a quadratic equation can have two roots, and a cubic equation can have three roots.