Use The Given Polynomial Function To Identify The Zeros Of The Function And The Multiplicity Of Each Zero.${ F(x) = 11(x+8) 3(x+4)(x+5) 2 }$[ \begin{tabular}{|c|c|} \hline Zeros & Mult. \ \hline □ \square □ & □ \square □
Understanding the Problem
To identify the zeros of a polynomial function and their corresponding multiplicities, we need to analyze the given function and apply the appropriate mathematical techniques. In this case, we are given the polynomial function:
Our goal is to find the zeros of this function, which are the values of x that make the function equal to zero, and determine the multiplicity of each zero.
What are Zeros and Multiplicity?
Before we dive into the solution, let's briefly discuss what zeros and multiplicity mean in the context of polynomial functions.
- Zeros: A zero of a polynomial function is a value of x that makes the function equal to zero. In other words, if f(x) = 0, then x is a zero of the function.
- Multiplicity: The multiplicity of a zero is the number of times the factor corresponding to that zero appears in the polynomial function. In other words, if a factor (x - a) appears k times in the polynomial function, then the multiplicity of the zero a is k.
Factoring the Polynomial Function
To identify the zeros of the polynomial function, we need to factor the function into its prime factors. In this case, the function is already factored as:
Identifying the Zeros
Now that we have factored the polynomial function, we can identify the zeros by setting each factor equal to zero and solving for x.
- Zero 1: Setting the first factor equal to zero, we get:
Solving for x, we get:
The multiplicity of this zero is 3, since the factor (x + 8) appears three times in the polynomial function.
- Zero 2: Setting the second factor equal to zero, we get:
Solving for x, we get:
The multiplicity of this zero is 1, since the factor (x + 4) appears only once in the polynomial function.
- Zero 3: Setting the third factor equal to zero, we get:
Solving for x, we get:
The multiplicity of this zero is 2, since the factor (x + 5) appears twice in the polynomial function.
Conclusion
In conclusion, we have identified the zeros of the polynomial function f(x) = 11(x+8)3(x+4)(x+5)2 as x = -8, x = -4, and x = -5, with multiplicities of 3, 1, and 2, respectively.
Tabular Representation of Zeros and Multiplicity
| Zeros | Mult. |
|---|---|
| -8 | 3 |
| -4 | 1 |
| -5 | 2 |
Discussion and Analysis
The zeros of a polynomial function are the values of x that make the function equal to zero. The multiplicity of each zero is the number of times the factor corresponding to that zero appears in the polynomial function.
In this case, we have identified the zeros of the polynomial function f(x) = 11(x+8)3(x+4)(x+5)2 as x = -8, x = -4, and x = -5, with multiplicities of 3, 1, and 2, respectively.
The multiplicity of each zero is an important concept in algebra, as it helps us understand the behavior of the polynomial function near each zero. For example, if a zero has a high multiplicity, it means that the function will have a more pronounced "bump" or "dip" near that zero.
Real-World Applications
The concept of zeros and multiplicity has many real-world applications in fields such as engineering, physics, and economics.
For example, in engineering, the zeros of a transfer function can be used to design filters and control systems. In physics, the zeros of a wave function can be used to describe the behavior of particles in a quantum system. In economics, the zeros of a demand function can be used to model the behavior of consumers in a market.
Conclusion
In conclusion, the zeros of a polynomial function are the values of x that make the function equal to zero, and the multiplicity of each zero is the number of times the factor corresponding to that zero appears in the polynomial function. By identifying the zeros and their multiplicities, we can gain a deeper understanding of the behavior of the polynomial function and its applications in real-world problems.
Q: What is the difference between a zero and a root of a polynomial function?
A: In the context of polynomial functions, the terms "zero" and "root" are often used interchangeably. However, some mathematicians make a distinction between the two terms. A zero of a polynomial function is a value of x that makes the function equal to zero, whereas a root of a polynomial function is a value of x that makes the polynomial equal to zero. In other words, a zero is a value of x that satisfies the equation f(x) = 0, whereas a root is a value of x that satisfies the equation f(x) = 0, where f(x) is the polynomial function.
Q: How do I find the zeros of a polynomial function?
A: To find the zeros of a polynomial function, you need to factor the function into its prime factors and set each factor equal to zero. Then, solve for x to find the values of x that make the function equal to zero.
Q: What is the multiplicity of a zero?
A: The multiplicity of a zero is the number of times the factor corresponding to that zero appears in the polynomial function. In other words, if a factor (x - a) appears k times in the polynomial function, then the multiplicity of the zero a is k.
Q: How do I determine the multiplicity of a zero?
A: To determine the multiplicity of a zero, you need to factor the polynomial function into its prime factors and count the number of times each factor appears. Then, identify the factor that corresponds to the zero you are interested in and count the number of times it appears.
Q: What is the relationship between the zeros and the multiplicity of a polynomial function?
A: The zeros and the multiplicity of a polynomial function are related in the sense that the multiplicity of a zero determines the behavior of the function near that zero. For example, if a zero has a high multiplicity, it means that the function will have a more pronounced "bump" or "dip" near that zero.
Q: Can a polynomial function have multiple zeros with the same multiplicity?
A: Yes, a polynomial function can have multiple zeros with the same multiplicity. For example, the polynomial function f(x) = (x - 2)^3(x - 3)^2 has two zeros, x = 2 and x = 3, both with a multiplicity of 3.
Q: How do I graph a polynomial function with multiple zeros?
A: To graph a polynomial function with multiple zeros, you need to identify the zeros and their multiplicities, and then use this information to sketch the graph of the function. You can use the fact that the function will have a more pronounced "bump" or "dip" near each zero, and that the multiplicity of each zero determines the steepness of the graph near that zero.
Q: What is the significance of the zeros and multiplicity of a polynomial function in real-world applications?
A: The zeros and multiplicity of a polynomial function are significant in real-world applications because they help us understand the behavior of the function and its applications in various fields such as engineering, physics, and economics.
Q: Can a polynomial function have a zero with a negative multiplicity?
A: No, a polynomial function cannot have a zero with a negative multiplicity. The multiplicity of a zero is always a non-negative integer, and it represents the number of times the factor corresponding to that zero appears in the polynomial function.
Q: How do I determine the degree of a polynomial function?
A: To determine the degree of a polynomial function, you need to identify the highest power of x in the polynomial function. The degree of the polynomial function is the exponent of the highest power of x.
Q: What is the relationship between the degree of a polynomial function and its zeros?
A: The degree of a polynomial function and its zeros are related in the sense that the degree of the polynomial function determines the number of zeros it has. For example, a polynomial function of degree 2 can have at most two zeros, whereas a polynomial function of degree 3 can have at most three zeros.
Q: Can a polynomial function have a zero that is not a real number?
A: Yes, a polynomial function can have a zero that is not a real number. For example, the polynomial function f(x) = x^2 + 1 has a zero at x = i, where i is the imaginary unit.
Q: How do I determine the zeros of a polynomial function with complex coefficients?
A: To determine the zeros of a polynomial function with complex coefficients, you need to use the quadratic formula and the properties of complex numbers. You can also use the fact that the zeros of a polynomial function with complex coefficients come in conjugate pairs.
Q: What is the significance of the zeros of a polynomial function with complex coefficients in real-world applications?
A: The zeros of a polynomial function with complex coefficients are significant in real-world applications because they help us understand the behavior of the function and its applications in various fields such as engineering, physics, and economics.
Q: Can a polynomial function have a zero that is a complex number with a negative imaginary part?
A: Yes, a polynomial function can have a zero that is a complex number with a negative imaginary part. For example, the polynomial function f(x) = x^2 + 1 has a zero at x = -i, where i is the imaginary unit.
Q: How do I determine the multiplicity of a zero of a polynomial function with complex coefficients?
A: To determine the multiplicity of a zero of a polynomial function with complex coefficients, you need to use the properties of complex numbers and the fact that the multiplicity of a zero is always a non-negative integer.
Q: What is the relationship between the multiplicity of a zero and the behavior of a polynomial function near that zero?
A: The multiplicity of a zero and the behavior of a polynomial function near that zero are related in the sense that the multiplicity of a zero determines the steepness of the graph of the function near that zero. For example, if a zero has a high multiplicity, it means that the function will have a more pronounced "bump" or "dip" near that zero.