Repeated Zero

Definition

A repeated zero is a root or solution of a polynomial equation that occurs more than once. It is a value of the independent variable that causes the polynomial function to evaluate to zero, and this value appears multiple times in the factored form of the polynomial.

5 Must Know Facts For Your Next Test

1. The multiplicity of a repeated zero is the number of times that zero appears as a factor in the factored form of the polynomial.
2. Repeated zeros affect the behavior of the polynomial function, such as the shape of the graph and the number of times the function crosses the x-axis.
3. Repeated zeros also impact the continuity and differentiability of the polynomial function, as they can create points of non-differentiability.
4. The degree of the polynomial is reduced by the multiplicity of a repeated zero when factoring the polynomial.
5. Repeated zeros can be identified by examining the factored form of the polynomial or by analyzing the graph of the function.

Review Questions

• Explain how the multiplicity of a repeated zero affects the factored form of a polynomial.
  • The multiplicity of a repeated zero determines the number of times that zero appears as a factor in the factored form of the polynomial. For example, if a polynomial has a repeated zero with a multiplicity of 3, the factored form of the polynomial will include the factor $(x - 0)^3$, where the exponent 3 represents the multiplicity of the repeated zero. The degree of the polynomial is reduced by the multiplicity of the repeated zero when factoring the expression.
• Describe the impact of repeated zeros on the behavior of a polynomial function.
  • Repeated zeros affect the behavior of a polynomial function in several ways. First, they influence the shape of the graph, as the function will cross the x-axis multiple times at the location of the repeated zero. Additionally, repeated zeros can create points of non-differentiability, as the function may not be continuous or differentiable at those points. The multiplicity of the repeated zero also determines the number of times the function touches or crosses the x-axis, which is an important characteristic to consider when analyzing the properties of the polynomial.
• Analyze how the presence of a repeated zero in a polynomial function relates to the factorization of the polynomial.
  • When a polynomial function has a repeated zero, the factored form of the polynomial will include that zero as a factor raised to a power equal to the multiplicity of the root. This means that the degree of the polynomial is reduced by the multiplicity of the repeated zero. For example, if a polynomial has a repeated zero with a multiplicity of 2, the factored form of the polynomial will include the factor $(x - 0)^2$, where the exponent 2 represents the multiplicity of the repeated zero. Understanding the relationship between repeated zeros and the factorization of polynomials is crucial for analyzing the properties and behavior of polynomial functions.

Related terms

Root:

A root of a polynomial function is a value of the independent variable that causes the function to evaluate to zero, making the function equal to zero at that point.

Multiplicity:

The multiplicity of a root is the number of times that root appears as a factor in the factored form of the polynomial.

Factored Form: The factored form of a polynomial is the expression that shows the polynomial as a product of linear factors, where each factor represents a root of the polynomial.