Root of A Polynomial Calculator with Multiplicitys

What is a Polynomial Root?

A polynomial root is a value of x 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.

Polynomial roots can be found using various methods, including factoring, completing the square, using the quadratic formula, and numerical methods for higher-degree polynomials.

Root Multiplicity

Root multiplicity refers to the number of times a root appears in the factorization of a polynomial. A root with multiplicity n means the polynomial has a factor of (x - r)ⁿ.

For example, in the polynomial (x - 2)³(x + 1), the root x = 2 has multiplicity 3, and x = -1 has multiplicity 1.

How to Find Polynomial Roots

Factoring

For lower-degree polynomials, factoring is often the simplest method. For example, to solve x² - 5x + 6 = 0, factor it as (x - 2)(x - 3) = 0, giving roots x = 2 and x = 3.

Quadratic Formula

For quadratic equations (degree 2), use the quadratic formula: x = [-b ± √(b² - 4ac)] / (2a).

Numerical Methods

For higher-degree polynomials, numerical methods like Newton's method or the bisection method can approximate roots.

Graphical Methods

Plotting the polynomial can help identify approximate locations of roots where the graph crosses the x-axis.

Using the Calculator

Our calculator finds the roots of a polynomial and determines their multiplicities. Enter your polynomial coefficients, and the calculator will display the roots and their multiplicities.

For example, if you enter the polynomial x³ - 6x² + 11x - 6, the calculator will show that the roots are x = 1, x = 2, and x = 3, each with multiplicity 1.