Finding the Zeroes of a Polynomial Function

Oct 13, 2025 by windailys.com 50 views

Introduction

In mathematics, a polynomial function is a function that can be expressed as a sum of terms, where each term is a product of a constant and a variable raised to a non-negative integer power. The zeroes of a polynomial function are the values of the variable that make the function equal to zero. In this article, we will explore the concept of zeroes of a polynomial function and determine which of the given options describes the zeroes of the graph of $f(x) = -x^5 + 9x^4 - 18x^3$ .

What are Zeroes?

The zeroes of a polynomial function are the values of the variable that make the function equal to zero. In other words, if $f(x) = 0$ , then $x$ is a zero of the function. The zeroes of a polynomial function can be real or complex numbers.

Types of Zeroes

There are two types of zeroes: real zeroes and complex zeroes. Real zeroes are the values of the variable that make the function equal to zero, while complex zeroes are the values of the variable that make the function equal to zero, but are not real numbers.

Multiplicity of Zeroes

The multiplicity of a zero is the number of times the factor $(x - r)$ appears in the factorization of the polynomial. In other words, if the polynomial can be factored as $(x - r)^n \cdot g(x)$ , where $g(x)$ is a polynomial and $n$ is a positive integer, then $r$ is a zero of multiplicity $n$ .

Finding the Zeroes of a Polynomial Function

To find the zeroes of a polynomial function, we can use various methods, such as factoring, synthetic division, or the rational root theorem. In this article, we will use the factoring method to find the zeroes of the polynomial function $f(x) = -x^5 + 9x^4 - 18x^3$ .

Factoring the Polynomial Function

To factor the polynomial function $f(x) = -x^5 + 9x^4 - 18x^3$ , we can start by factoring out the greatest common factor, which is $-x^3$ . This gives us:

f(x) = -x^3(x^2 - 9x + 18)

Next, we can factor the quadratic expression $x^2 - 9x + 18$ as $(x - 3)(x - 6)$ . This gives us:

f(x) = -x^3(x - 3)(x - 6)

Finding the Zeroes

Now that we have factored the polynomial function, we can find the zeroes by setting each factor equal to zero. This gives us:

-x^3 = 0 \Rightarrow x = 0

(x - 3) = 0 \Rightarrow x = 3

(x - 6) = 0 \Rightarrow x = 6

Determining the Multiplicity of the Zeroes

To determine the multiplicity of the zeroes, we can look at the factorization of the polynomial function. We can see that the factor $-x^3$ has a multiplicity of 3, while the factors $(x - 3)$ and $(x - 6)$ have a multiplicity of 1.

Conclusion

In conclusion, the zeroes of the graph of $f(x) = -x^5 + 9x^4 - 18x^3$ are 0 with multiplicity 3, 3 with multiplicity 1, and 6 with multiplicity 1. This is option B.

Answer

The correct answer is B. 0 with multiplicity 3, 3 with multiplicity 1, and 6 with multiplicity 1.

Discussion

This problem requires the student to understand the concept of zeroes of a polynomial function and to be able to find the zeroes of a given polynomial function. The student must also be able to determine the multiplicity of the zeroes. This problem is relevant to the topic of algebra and is a good example of how to use factoring to find the zeroes of a polynomial function.

Additional Resources

For additional resources on this topic, please see the following:

Khan Academy: Polynomial Functions
Mathway: Factoring Polynomials
Wolfram Alpha: Polynomial Factorization
Q&A: Understanding the Zeroes of a Polynomial Function =====================================================

Q: What are the zeroes of a polynomial function?

A: The zeroes of a polynomial function are the values of the variable that make the function equal to zero. In other words, if $f(x) = 0$ , then $x$ is a zero of the function.

Q: What are the different types of zeroes?

A: There are two types of zeroes: real zeroes and complex zeroes. Real zeroes are the values of the variable that make the function equal to zero, while complex zeroes are the values of the variable that make the function equal to zero, but are not real numbers.

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

A: The multiplicity of a zero is the number of times the factor $(x - r)$ appears in the factorization of the polynomial. In other words, if the polynomial can be factored as $(x - r)^n \cdot g(x)$ , where $g(x)$ is a polynomial and $n$ is a positive integer, then $r$ is a zero of multiplicity $n$ .

Q: How do you find the zeroes of a polynomial function?

A: To find the zeroes of a polynomial function, you can use various methods, such as factoring, synthetic division, or the rational root theorem. In this article, we used the factoring method to find the zeroes of the polynomial function $f(x) = -x^5 + 9x^4 - 18x^3$ .

Q: What is the difference between a zero and a root?

A: A zero and a root are the same thing. The term "zero" is often used in the context of polynomial functions, while the term "root" is often used in the context of equations.

Q: Can a polynomial function have multiple zeroes?

A: Yes, a polynomial function can have multiple zeroes. In fact, a polynomial function can have any number of zeroes, including zero, one, two, three, and so on.

Q: How do you determine the number of zeroes of a polynomial function?

A: To determine the number of zeroes of a polynomial function, you can use the fundamental theorem of algebra, which states that a polynomial function of degree $n$ has exactly $n$ complex zeroes, counting multiplicities.

Q: What is the significance of the zeroes of a polynomial function?

A: The zeroes of a polynomial function are significant because they can be used to determine the behavior of the function. For example, if a polynomial function has a zero at $x = a$ , then the function will be equal to zero at $x = a$ . This can be useful in a variety of applications, such as physics and engineering.

Q: Can you give an example of a polynomial function with multiple zeroes?

A: Yes, here is an example of a polynomial function with multiple zeroes:

f(x) = (x - 2)^2(x - 3)(x - 4)

This polynomial function has four zeroes: $x = 2$ with multiplicity 2, $x = 3$ with multiplicity 1, and $x = 4$ with multiplicity 1.

Q: How do you find the zeroes of a polynomial function with multiple zeroes?

A: To find the zeroes of a polynomial function with multiple zeroes, you can use the factoring method, synthetic division, or the rational root theorem. In this article, we used the factoring method to find the zeroes of the polynomial function $f(x) = -x^5 + 9x^4 - 18x^3$ .

Q: What is the relationship between the zeroes of a polynomial function and its graph?

A: The zeroes of a polynomial function are the points on the graph where the function intersects the x-axis. In other words, if a polynomial function has a zero at $x = a$ , then the graph of the function will intersect the x-axis at $x = a$ .

Q: Can you give an example of a polynomial function with no zeroes?

A: Yes, here is an example of a polynomial function with no zeroes:

f(x) = x^2 + 1

This polynomial function has no zeroes because it is always positive for all real values of $x$ .

Q: How do you determine if a polynomial function has any zeroes?

A: To determine if a polynomial function has any zeroes, you can use the rational root theorem or the fundamental theorem of algebra. If the polynomial function has any rational roots, then it will have at least one zero. If the polynomial function has no rational roots, then it may have complex zeroes.

Q: What is the significance of the zeroes of a polynomial function in real-world applications?

A: The zeroes of a polynomial function are significant in real-world applications because they can be used to determine the behavior of the function. For example, in physics and engineering, the zeroes of a polynomial function can be used to determine the stability of a system or the behavior of a physical system.

Q: Can you give an example of a polynomial function with complex zeroes?

A: Yes, here is an example of a polynomial function with complex zeroes:

f(x) = x^2 + 1

This polynomial function has two complex zeroes: $x = i$ and $x = -i$ .

Q: How do you find the complex zeroes of a polynomial function?

A: To find the complex zeroes of a polynomial function, you can use the factoring method, synthetic division, or the rational root theorem. In this article, we used the factoring method to find the complex zeroes of the polynomial function $f(x) = -x^5 + 9x^4 - 18x^3$ .

Q: What is the relationship between the complex zeroes of a polynomial function and its graph?

A: The complex zeroes of a polynomial function are the points on the graph where the function intersects the imaginary axis. In other words, if a polynomial function has a complex zero at $x = a + bi$ , then the graph of the function will intersect the imaginary axis at $x = a + bi$ .