How to Find the Zeroes of a Polynomial Function

Oct 13, 2025 by miodeya.com 39 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 the correct description of the zeroes of the graph of $f(x)=-x^5+9x^4-18x^3$ .

What are Zeroes of a Polynomial Function?

The zeroes of a polynomial function are the values of the variable that make the function equal to zero. In other words, if we have a polynomial function $f(x)$ , then the zeroes of $f(x)$ are the values of $x$ such that $f(x) = 0$ . For example, if we have the polynomial function $f(x) = x^2 - 4$ , then the zeroes of $f(x)$ are $x = 2$ and $x = -2$ , because $f(2) = 2^2 - 4 = 0$ and $f(-2) = (-2)^2 - 4 = 0$ .

How to Find the Zeroes of a Polynomial Function

There are several methods to find the zeroes of a polynomial function, including factoring, synthetic division, and the rational root theorem. Factoring involves expressing the polynomial function as a product of simpler polynomials, while synthetic division involves dividing the polynomial function by a linear factor. The rational root theorem states that if a rational number $p/q$ is a root of the polynomial function, then $p$ must be a factor of the constant term and $q$ must be a factor of the leading coefficient.

The Graph of $f(x)=-x^5+9x^4-18x^3$

The graph of $f(x)=-x^5+9x^4-18x^3$ is a polynomial function of degree 5. To find the zeroes of this function, we can use the fact that the zeroes of a polynomial function are the values of the variable that make the function equal to zero. We can start by factoring the polynomial function:

f(x) = -x^5 + 9x^4 - 18x^3 = -x^3(x^2 - 9x + 18)

We can see that $x = 0$ is a zero of the function, because $f(0) = 0$ . We can also see that $x^2 - 9x + 18$ is a quadratic factor of the function.

Finding the Zeroes of the Quadratic Factor

To find the zeroes of the quadratic factor $x^2 - 9x + 18$ , we can use the quadratic formula:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

In this case, $a = 1$ , $b = -9$ , and $c = 18$ . Plugging these values into the quadratic formula, we get:

x = \frac{-(-9) \pm \sqrt{(-9)^2 - 4(1)(18)}}{2(1)}

Simplifying, we get:

x = \frac{9 \pm \sqrt{81 - 72}}{2}

x = \frac{9 \pm \sqrt{9}}{2}

x = \frac{9 \pm 3}{2}

Therefore, the zeroes of the quadratic factor are $x = 6$ and $x = 3$ .

Conclusion

In conclusion, the zeroes of the graph of $f(x)=-x^5+9x^4-18x^3$ are $x = 0$ with multiplicity 3, $x = 3$ with multiplicity 1, and $x = 6$ with multiplicity 1. This is because $x = 0$ is a zero of the function with multiplicity 3, and $x = 3$ and $x = 6$ are zeroes of the quadratic factor with multiplicity 1.

Answer

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 we have a polynomial function $f(x)$ , then the zeroes of $f(x)$ are the values of $x$ such that $f(x) = 0$ .

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

A: There are several methods to find the zeroes of a polynomial function, including factoring, synthetic division, and the rational root theorem. Factoring involves expressing the polynomial function as a product of simpler polynomials, while synthetic division involves dividing the polynomial function by a linear factor. The rational root theorem states that if a rational number $p/q$ is a root of the polynomial function, then $p$ must be a factor of the constant term and $q$ must be a factor of the leading coefficient.

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

A: A zero of a polynomial function is a value of the variable that makes the function equal to zero, while a root of a polynomial function is a value of the variable that makes the function equal to zero, and also satisfies the equation $f(x) = 0$ . In other words, a zero is a value of the variable that makes the function equal to zero, while a root is a value of the variable that makes the function equal to zero and also satisfies the equation $f(x) = 0$ .

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

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

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

A: The zeroes of a polynomial function are significant because they determine the behavior of the function. For example, if a polynomial function has a zero at $x = r$ , then the function will change sign at $x = r$ . This means that the function will have a local maximum or minimum at $x = r$ , depending on the sign of the function at $x = r$ .

Q: How do I use the zeroes of a polynomial function to graph the function?

A: To graph a polynomial function, we can use the zeroes of the function to determine the x-intercepts of the graph. We can then use the x-intercepts to draw the graph of the function. For example, if a polynomial function has zeroes at $x = r_1, r_2, \ldots, r_n$ , then the graph of the function will have x-intercepts at $x = r_1, r_2, \ldots, r_n$ .

Q: What are some common mistakes to avoid when finding the zeroes of a polynomial function?

A: Some common mistakes to avoid when finding the zeroes of a polynomial function include:

Not factoring the polynomial function correctly
Not using the rational root theorem correctly
Not using synthetic division correctly
Not checking for extraneous solutions
Not using the correct method to find the zeroes of the polynomial function

Q: How do I check my work when finding the zeroes of a polynomial function?

A: To check your work when finding the zeroes of a polynomial function, you can use the following steps:

Check that the polynomial function is factored correctly
Check that the rational root theorem is used correctly
Check that synthetic division is used correctly
Check that the correct method is used to find the zeroes of the polynomial function
Check that the zeroes are not extraneous solutions

Q: What are some real-world applications of the zeroes of a polynomial function?

A: Some real-world applications of the zeroes of a polynomial function include:

Modeling population growth
Modeling chemical reactions
Modeling electrical circuits
Modeling mechanical systems
Modeling financial systems

Q: How do I use the zeroes of a polynomial function to solve real-world problems?

A: To use the zeroes of a polynomial function to solve real-world problems, you can follow these steps:

Identify the problem and determine the type of polynomial function that is needed to model the problem
Use the polynomial function to model the problem
Find the zeroes of the polynomial function
Use the zeroes to determine the behavior of the function
Use the behavior of the function to solve the problem

Conclusion

In conclusion, the zeroes of a polynomial function are an important concept in mathematics that has many real-world applications. By understanding the zeroes of a polynomial function, we can use the function to model real-world problems and solve them. We can also use the zeroes to determine the behavior of the function and make predictions about the behavior of the function.