Identify The Zeros, Their Multiplicity, And The Effect They Have On The Graph Of The Function. Type crosses Or bounces For Effect, And Type The Zeros In Order From Least To Greatest.Given The Function: $f(x) = X(2x-1)^2(x+8)^5$

Feb 25, 2025 by ADMIN 232 views

**Identifying Zeros and Their Multiplicity in a Polynomial Function**

Understanding the Concept of Zeros in a Polynomial Function

In mathematics, a zero of a function is a value of the input variable (x) that makes the function equal to zero. In other words, it is a solution to the equation f(x) = 0. The concept of zeros is crucial in understanding the behavior of a function, particularly in graphing and analyzing polynomial functions. In this article, we will focus on identifying the zeros, their multiplicity, and the effect they have on the graph of the given function: $f(x) = x(2x-1)^2(x+8)^5$ .

The Importance of Multiplicity in Zeros

Multiplicity refers to the number of times a zero occurs in a function. In other words, it is the number of times the function touches or crosses the x-axis at a particular point. Multiplicity is an essential concept in understanding the behavior of a function, particularly in graphing and analyzing polynomial functions. A zero with a multiplicity of 1 means that the function touches the x-axis at that point, while a zero with a multiplicity greater than 1 means that the function crosses the x-axis at that point.

Identifying Zeros in the Given Function

To identify the zeros of the given function, we need to set the function equal to zero and solve for x. The given function is $f(x) = x(2x-1)^2(x+8)^5$ . To find the zeros, we need to set each factor equal to zero and solve for x.

Solving for x in the First Factor

The first factor is x, which can be set equal to zero as follows:

$x = 0$

This means that the function touches the x-axis at x = 0.

Solving for x in the Second Factor

The second factor is $(2x-1)^2$ , which can be set equal to zero as follows:

$(2x-1)^2 = 0$

Solving for x, we get:

$2x - 1 = 0$

$2x = 1$

$x = \frac{1}{2}$

This means that the function touches the x-axis at x = 1/2.

Solving for x in the Third Factor

The third factor is $(x+8)^5$ , which can be set equal to zero as follows:

$(x+8)^5 = 0$

Solving for x, we get:

$x + 8 = 0$

$x = -8$

This means that the function crosses the x-axis at x = -8.

Determining the Effect of Zeros on the Graph

Now that we have identified the zeros of the given function, we need to determine the effect they have on the graph. The zeros of a function can affect the graph in several ways, including:

Crossing the x-axis: If a zero has a multiplicity of 1, the function will touch the x-axis at that point. If a zero has a multiplicity greater than 1, the function will cross the x-axis at that point.
Touching the x-axis: If a zero has a multiplicity of 1, the function will touch the x-axis at that point.
Not affecting the graph: If a zero has a multiplicity of 0, the function will not be affected by that zero.

In the case of the given function, the zeros are x = 0, x = 1/2, and x = -8. The effect of these zeros on the graph is as follows:

x = 0: The function touches the x-axis at x = 0.
x = 1/2: The function touches the x-axis at x = 1/2.
x = -8: The function crosses the x-axis at x = -8.

Conclusion

In conclusion, identifying the zeros, their multiplicity, and the effect they have on the graph of a function is crucial in understanding the behavior of the function. The zeros of a function can affect the graph in several ways, including crossing the x-axis, touching the x-axis, and not affecting the graph. By identifying the zeros and their multiplicity, we can gain a deeper understanding of the behavior of a function and its graph.

Final Answer

The final answer is:

Zeros: x = 0, x = 1/2, and x = -8
Multiplicity: 1, 1, and 5
Effect on the graph: touches the x-axis at x = 0, touches the x-axis at x = 1/2, and crosses the x-axis at x = -8

Q: What is a zero of a function?

A: A zero of a function is a value of the input variable (x) that makes the function equal to zero. In other words, it is a solution to the equation f(x) = 0.

Q: Why is it important to identify the zeros of a function?

A: Identifying the zeros of a function is crucial in understanding the behavior of the function, particularly in graphing and analyzing polynomial functions. Zeros can affect the graph of a function in several ways, including crossing the x-axis, touching the x-axis, and not affecting the graph.

Q: How do I identify the zeros of a function?

A: To identify the zeros of a function, you need to set the function equal to zero and solve for x. You can do this by setting each factor of the function equal to zero and solving for x.

Q: What is multiplicity in the context of zeros?

A: Multiplicity refers to the number of times a zero occurs in a function. In other words, it is the number of times the function touches or crosses the x-axis at a particular point.

Q: How do I determine the multiplicity of a zero?

A: To determine the multiplicity of a zero, you need to examine the factor of the function that corresponds to the zero. If the factor is raised to a power greater than 1, the zero has a multiplicity greater than 1. If the factor is raised to a power of 1, the zero has a multiplicity of 1.

Q: What is the effect of a zero on the graph of a function?

A: The effect of a zero on the graph of a function depends on its multiplicity. If a zero has a multiplicity of 1, the function touches the x-axis at that point. If a zero has a multiplicity greater than 1, the function crosses the x-axis at that point.

Q: Can a zero have a multiplicity of 0?

A: Yes, a zero can have a multiplicity of 0. This means that the function does not touch or cross the x-axis at that point.

Q: How do I determine the effect of a zero on the graph of a function?

A: To determine the effect of a zero on the graph of a function, you need to examine the multiplicity of the zero. If the multiplicity is 1, the function touches the x-axis at that point. If the multiplicity is greater than 1, the function crosses the x-axis at that point.

Q: Can I use technology to help me identify the zeros of a function?

A: Yes, you can use technology such as graphing calculators or computer software to help you identify the zeros of a function. These tools can graph the function and help you identify the zeros.

Q: How do I use technology to help me identify the zeros of a function?

A: To use technology to help you identify the zeros of a function, you need to enter the function into the graphing calculator or computer software and graph the function. The graph will show you the zeros of the function.

Q: What are some common mistakes to avoid when identifying the zeros of a function?

A: Some common mistakes to avoid when identifying the zeros of a function include:

Not setting the function equal to zero: Make sure to set the function equal to zero before solving for x.
Not solving for x: Make sure to solve for x after setting the function equal to zero.
Not examining the multiplicity of the zero: Make sure to examine the multiplicity of the zero to determine its effect on the graph of the function.

Q: How can I practice identifying the zeros of a function?

A: You can practice identifying the zeros of a function by working through examples and exercises in a textbook or online resource. You can also use technology such as graphing calculators or computer software to help you identify the zeros of a function.

Q: What are some real-world applications of identifying the zeros of a function?

A: Identifying the zeros of a function has many real-world applications, including:

Physics: Identifying the zeros of a function can help you understand the behavior of physical systems, such as the motion of objects.
Engineering: Identifying the zeros of a function can help you design and optimize systems, such as electrical circuits.
Economics: Identifying the zeros of a function can help you understand the behavior of economic systems, such as the behavior of supply and demand.

Q: How can I use identifying the zeros of a function to solve real-world problems?

A: You can use identifying the zeros of a function to solve real-world problems by applying the concepts and techniques you have learned to real-world scenarios. For example, you can use identifying the zeros of a function to understand the behavior of physical systems, design and optimize systems, and understand the behavior of economic systems.