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Multiplicity of Root Calculator

Reviewed by Calculator Editorial Team

Understanding the multiplicity of roots in polynomials is essential for analyzing their behavior and solving equations. This calculator helps you determine how many times each root appears in a polynomial equation.

What is Multiplicity of Root?

The multiplicity of a root in a polynomial equation refers to the number of times that root appears as a solution to the equation. For example, in the equation (x-2)³(x+1)²=0, the root x=2 has a multiplicity of 3, and x=-1 has a multiplicity of 2.

Multiplicity affects the behavior of the polynomial graph near the root. Higher multiplicity roots create flatter "bumps" or "dips" in the graph, while lower multiplicity roots create sharper turns.

Note: Multiplicity is always a positive integer. A root with multiplicity 1 is called a simple root, while roots with higher multiplicity are called multiple roots.

How to Calculate Multiplicity of Root

To calculate the multiplicity of a root, you can use the following steps:

  1. Factor the polynomial completely.
  2. Identify all the roots (solutions to the equation).
  3. Count how many times each root appears in the factored form.
For a polynomial P(x) = a(x-r)ⁿQ(x), where Q(x) does not have r as a root, the multiplicity of root r is n.

For example, consider the polynomial P(x) = (x-3)²(x+1)³. Here:

  • The root x=3 has multiplicity 2.
  • The root x=-1 has multiplicity 3.

Step-by-Step Example

Let's find the multiplicity of roots for P(x) = x³ - 6x² + 11x - 6.

  1. Factor the polynomial: P(x) = (x-1)(x-2)(x-3).
  2. Identify the roots: x=1, x=2, x=3.
  3. Count the multiplicity: Each root appears once, so each has multiplicity 1.

Example Calculations

Here are some example calculations of root multiplicity:

Polynomial Roots Multiplicity
x² - 4 x=2, x=-2 1 for each
(x-1)³ x=1 3
x³ - 3x² + 2x x=0, x=1, x=2 1 for each
(x+2)²(x-3) x=-2, x=3 2 for x=-2, 1 for x=3

These examples demonstrate how different polynomials can have roots with varying multiplicities.

FAQ

What is the difference between simple and multiple roots?
A simple root has multiplicity 1, meaning it appears once in the factored form of the polynomial. A multiple root has multiplicity greater than 1, meaning it appears more than once.
How does multiplicity affect the graph of a polynomial?
Higher multiplicity roots create flatter "bumps" or "dips" in the graph, while lower multiplicity roots create sharper turns. The graph touches or crosses the x-axis based on the multiplicity.
Can a polynomial have more than one root with the same multiplicity?
Yes, a polynomial can have multiple roots with the same multiplicity. For example, (x-1)²(x-2)² has two roots, each with multiplicity 2.