key term - Simple Zeros

Definition

A simple zero of a polynomial function is a real number that makes the function equal to zero and has a multiplicity of one. In other words, it is a point where the graph of the polynomial function crosses the x-axis and the function changes sign at that point.

5 Must Know Facts For Your Next Test

1. The number of simple zeros of a polynomial function is equal to the degree of the polynomial, provided that the polynomial has no repeated roots.
2. Simple zeros are important in determining the behavior of the graph of a polynomial function, as they represent the points where the function changes sign.
3. The presence of simple zeros affects the number of turning points (local maxima and minima) and the number of intervals where the function is increasing or decreasing.
4. Finding the simple zeros of a polynomial function is a crucial step in sketching the graph of the function, as they provide valuable information about the function's behavior.
5. Simple zeros can be found using various methods, such as factoring, the quadratic formula, or numerical approximation techniques.

Review Questions

• Explain the significance of simple zeros in the context of graphing polynomial functions.
  • Simple zeros are critical points on the graph of a polynomial function, as they represent the points where the function changes sign and crosses the x-axis. The number of simple zeros is equal to the degree of the polynomial, and their location determines the number of turning points and the intervals where the function is increasing or decreasing. Understanding the properties of simple zeros is essential for sketching the graph of a polynomial function and analyzing its behavior.
• Describe the relationship between the multiplicity of a zero and its impact on the graph of a polynomial function.
  • The multiplicity of a zero refers to the number of times the zero is repeated as a root of the polynomial function. A simple zero has a multiplicity of one, meaning it is a single root that causes the graph to cross the x-axis. In contrast, a multiple zero with a multiplicity greater than one results in the graph touching the x-axis without crossing it. This difference in behavior affects the shape and characteristics of the polynomial function's graph, such as the number of turning points and the intervals of increase and decrease.
• Analyze how the presence of simple zeros influences the overall behavior and properties of a polynomial function's graph.
  • The presence of simple zeros has a significant impact on the graph of a polynomial function. Simple zeros represent points where the function changes sign, dividing the graph into intervals where the function is either positive or negative. This, in turn, affects the number and location of turning points (local maxima and minima) on the graph. Additionally, the simple zeros determine the number of times the graph crosses the x-axis, which is crucial for understanding the function's behavior and sketching its graph accurately. The properties of simple zeros, such as their multiplicity and location, are essential for analyzing the overall shape, symmetry, and other characteristics of the polynomial function's graph.

Related terms

Multiplicity:

The multiplicity of a zero refers to the number of times the zero is repeated as a root of the polynomial function. A simple zero has a multiplicity of one.

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 one or more variables raised to a nonnegative integer power.

Graphing Polynomial Functions: The graph of a polynomial function is a smooth curve that may have one or more points where the curve crosses the x-axis, known as the zeros or roots of the function.