Multiplicity

Definition

Multiplicity refers to the number of times a particular value occurs as a zero or root of a polynomial function. It describes the order or repetition of a zero or root within the function's graph or equation.

5 Must Know Facts For Your Next Test

1. The multiplicity of a zero or root determines the behavior of the polynomial function's graph near that point.
2. A zero or root with multiplicity 1 is called a simple zero or root, and the graph will have a single intersection with the x-axis at that point.
3. A zero or root with multiplicity greater than 1 is called a repeated zero or root, and the graph will have a higher-order contact with the x-axis at that point.
4. The degree of the polynomial function is equal to the sum of the multiplicities of all its zeros or roots.
5. Multiplicity plays a crucial role in determining the factorization of a polynomial function and the behavior of its graph.

Review Questions

• Explain how the multiplicity of a zero or root affects the graph of a polynomial function.
  • The multiplicity of a zero or root determines the behavior of the polynomial function's graph near that point. A zero or root with multiplicity 1 is a simple zero or root, and the graph will have a single intersection with the x-axis at that point. However, a zero or root with multiplicity greater than 1 is a repeated zero or root, and the graph will have a higher-order contact with the x-axis at that point. This means the graph will either have a sharp corner or a flat spot at the repeated zero or root, depending on the specific multiplicity.
• Describe the relationship between the degree of a polynomial function and the sum of the multiplicities of its zeros or roots.
  • The degree of a polynomial function is equal to the sum of the multiplicities of all its zeros or roots. This means that the degree of the polynomial is determined by the number and repetition of its zeros or roots. For example, a polynomial function of degree 4 could have four simple zeros (multiplicity 1 each), two repeated zeros with multiplicity 2 each, or any other combination of zeros or roots that add up to a total multiplicity of 4.
• Analyze how multiplicity affects the factorization and behavior of a polynomial function.
  • Multiplicity plays a crucial role in determining the factorization of a polynomial function. Repeated zeros or roots will result in the function being expressed as a product of linear factors raised to powers equal to their respective multiplicities. This factored form provides important insights into the behavior of the polynomial function's graph. The multiplicity of a zero or root determines the order of contact between the graph and the x-axis, which in turn affects the function's local behavior, such as the presence of sharp corners or flat spots on the graph.

Related terms

Zero:

A zero of a polynomial function is a value of the input variable that makes the function equal to zero.

Root:

A root of a polynomial function is a value of the input variable that makes the function equal to zero, similar to a zero.

Polynomial Function:

A polynomial function is a function that can be expressed as the sum of a finite number of nonzero terms, each of which is a product of a constant and a variable raised to a nonnegative integer power.