Single Zero

Definition

A single zero of a polynomial function is a real number that, when substituted into the function, results in the function evaluating to zero. This means that the graph of the polynomial function will intersect the x-axis at that point, indicating a point where the function changes from positive to negative or vice versa.

5 Must Know Facts For Your Next Test

1. A single zero of a polynomial function is a real number that makes the function equal to zero when substituted into the function.
2. The graph of a polynomial function with a single zero will intersect the x-axis at that point, indicating a change in the sign of the function.
3. Identifying the single zeros of a polynomial function is important for understanding the behavior and properties of the function, such as its domain, range, and end behavior.
4. The single zeros of a polynomial function can be found by factoring the function or using other algebraic techniques, such as the Rational Root Theorem.
5. The multiplicity of a single zero refers to the number of times the zero appears as a factor in the factorization of the polynomial function.

Review Questions

• Explain how a single zero of a polynomial function is represented on the graph of the function.
  • A single zero of a polynomial function is represented on the graph of the function as a point where the graph intersects the x-axis. At this point, the function changes from positive to negative or vice versa, indicating that the function evaluates to zero at that specific x-value. The graph will cross the x-axis at the single zero, and the point of intersection represents the value of the single zero.
• Describe the relationship between the factorization of a polynomial function and its single zeros.
  • The factorization of a polynomial function is closely related to its single zeros. By expressing the polynomial as a product of simpler polynomial functions, the single zeros of the original function can be identified as the roots or x-intercepts of the factored form. The multiplicity of a single zero is determined by the number of times that factor appears in the factorization. Understanding the factorization of a polynomial function can provide valuable insights into its single zeros and their properties.
• Analyze the significance of identifying the single zeros of a polynomial function in the context of understanding its behavior and properties.
  • Identifying the single zeros of a polynomial function is crucial for understanding its behavior and properties. The single zeros represent the values of the variable that make the function equal to zero, which correspond to the x-intercepts of the graph. These points of intersection with the x-axis are important in determining the function's domain, range, and end behavior. Additionally, the multiplicity of the single zeros can provide information about the function's behavior, such as whether it changes from increasing to decreasing or vice versa. By analyzing the single zeros of a polynomial function, you can gain valuable insights into its overall characteristics and how it behaves within its domain.

Related terms

Polynomial Function: A polynomial function is a function that can be expressed as the sum of a finite number of non-zero terms, each of which is a product of a constant and one or more variables raised to a non-negative integer power.

Roots of a Polynomial: The roots of a polynomial function are the values of the variable that make the function equal to zero. These roots correspond to the x-intercepts of the graph of the polynomial function.

Factorization: Factorization is the process of expressing a polynomial function as a product of simpler polynomial functions, which can help in identifying the roots or zeros of the original function.