Multiple Zero

Definition

A multiple zero, also known as a repeated root or repeated zero, is a value of the independent variable for which a polynomial function evaluates to zero, and this value occurs more than once in the function's set of zeros. In other words, it is a point where the graph of the polynomial function touches or crosses the x-axis more than once.

5 Must Know Facts For Your Next Test

1. The presence of a multiple zero in the set of zeros for a polynomial function affects the shape and behavior of the graph of the function.
2. When a polynomial function has a multiple zero, the graph of the function will touch or cross the x-axis at that point more than once.
3. The multiplicity of a zero determines the number of times the graph of the function will touch or cross the x-axis at that point.
4. Multiple zeros can be identified through the factorization of the polynomial function, where repeated linear factors indicate the presence of multiple zeros.
5. The behavior of the function near a multiple zero is influenced by the multiplicity, with higher multiplicities leading to sharper, more pronounced behavior at the zero.

Review Questions

• Explain how the presence of a multiple zero affects the graph of a polynomial function.
  • The presence of a multiple zero in the set of zeros for a polynomial function affects the shape and behavior of the graph of the function. When a polynomial function has a multiple zero, the graph of the function will touch or cross the x-axis at that point more than once. The multiplicity of the zero, which is the number of times the zero occurs in the set of zeros, determines the number of times the graph will touch or cross the x-axis at that point. Higher multiplicities lead to sharper, more pronounced behavior at the zero, with the graph exhibiting a more pronounced cusp or spike at the multiple zero.
• Describe how the factorization of a polynomial function can be used to identify multiple zeros.
  • The factorization of a polynomial function can be used to identify the presence of multiple zeros. When a polynomial function is expressed as a product of linear factors, repeated linear factors indicate the presence of multiple zeros. For example, if a polynomial function can be written as $P(x) = (x - a)^m(x - b)(x - c)$, where $m > 1$, then the value $a$ is a multiple zero with a multiplicity of $m$. The factorization of the polynomial function, therefore, provides a way to identify the multiple zeros and their multiplicities, which is important for understanding the behavior of the function's graph.
• Analyze the impact of the multiplicity of a zero on the behavior of a polynomial function near that zero.
  • The multiplicity of a zero in a polynomial function has a significant impact on the behavior of the function near that zero. As the multiplicity increases, the behavior of the function near the zero becomes more pronounced and sharper. For example, if a polynomial function has a zero with multiplicity 1, the graph will simply cross the x-axis at that point. However, if the multiplicity is 2, the graph will touch the x-axis and have a local minimum or maximum at that point. With a multiplicity of 3 or higher, the graph will exhibit a more pronounced cusp or spike at the zero, with the function's behavior becoming increasingly sharper as the multiplicity increases. This is an important consideration when analyzing the properties and behavior of polynomial functions.

Related terms

Polynomial Function: A polynomial function is a function that can be expressed in the form $P(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$, where $a_n, a_{n-1}, ..., a_1, a_0$ are real numbers and $n$ is a non-negative integer.

Multiplicity: The multiplicity of a zero is the number of times that the zero occurs in the set of zeros for the polynomial function.

Factorization: Factorization is the process of expressing a polynomial function as a product of linear factors, which can be used to identify the zeros of the function.