Determining the Multiplicity of Each Zero

Oct 13, 2025 by miodeya.com 54 views

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Introduction

In mathematics, a polynomial function is a function of the form $G(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$ , where $a_n \neq 0$ and $n$ is a non-negative integer. The zeros of a polynomial function are the values of $x$ for which $G(x) = 0$ . In this article, we will discuss how to determine the zeros of a given polynomial function and the multiplicity of each zero. We will also list the zeros from smallest to largest.

Understanding the Problem

The given polynomial function is $G(x) = x^3 - 6x^2 + 11x - 6$ . To determine the zeros of this function, we need to find the values of $x$ for which $G(x) = 0$ . This means that we need to solve the equation $x^3 - 6x^2 + 11x - 6 = 0$ .

Factoring the Polynomial

To solve the equation $x^3 - 6x^2 + 11x - 6 = 0$ , we can try to factor the polynomial. Factoring a polynomial means expressing it as a product of simpler polynomials. In this case, we can factor the polynomial as follows:

x^3 - 6x^2 + 11x - 6 = (x - 1)(x - 2)(x - 3)

Finding the Zeros

Now that we have factored the polynomial, we can find the zeros of the function by setting each factor equal to zero and solving for $x$ . This gives us the following equations:

x - 1 = 0 \Rightarrow x = 1

x - 2 = 0 \Rightarrow x = 2

x - 3 = 0 \Rightarrow x = 3

Determining the Multiplicity of Each Zero

The multiplicity of a zero is the number of times that factor appears in the factored form of the polynomial. In this case, each factor appears only once, so the multiplicity of each zero is 1.

Listing the Zeros from Smallest to Largest

Now that we have found the zeros of the function and determined the multiplicity of each zero, we can list the zeros from smallest to largest. The zeros are $x = 1$ , $x = 2$ , and $x = 3$ .

Conclusion

In this article, we have discussed how to determine the zeros of a given polynomial function and the multiplicity of each zero. We have also listed the zeros from smallest to largest. The zeros of the polynomial function $G(x) = x^3 - 6x^2 + 11x - 6$ are $x = 1$ , $x = 2$ , and $x = 3$ , each with a multiplicity of 1.

Example 1: Finding Zeros of a Polynomial Function

Let's consider another example. Suppose we have the polynomial function $G(x) = x^2 + 4x + 4$ . To find the zeros of this function, we can try to factor the polynomial. Factoring the polynomial gives us:

x^2 + 4x + 4 = (x + 2)(x + 2)

Now that we have factored the polynomial, we can find the zeros of the function by setting each factor equal to zero and solving for $x$ . This gives us the following equation:

x + 2 = 0 \Rightarrow x = -2

The multiplicity of the zero $x = -2$ is 2, since the factor $(x + 2)$ appears twice in the factored form of the polynomial.

Example 2: Finding Zeros of a Polynomial Function with Complex Zeros

Let's consider another example. Suppose we have the polynomial function $G(x) = x^3 + 3x^2 + 3x + 1$ . To find the zeros of this function, we can try to factor the polynomial. Factoring the polynomial gives us:

x^3 + 3x^2 + 3x + 1 = (x + 1)^3

Now that we have factored the polynomial, we can find the zeros of the function by setting each factor equal to zero and solving for $x$ . This gives us the following equation:

(x + 1)^3 = 0 \Rightarrow x + 1 = 0 \Rightarrow x = -1

The multiplicity of the zero $x = -1$ is 3, since the factor $(x + 1)$ appears three times in the factored form of the polynomial.

Conclusion

Tips and Tricks

When factoring a polynomial, try to find the greatest common factor (GCF) of the terms.
When solving a polynomial equation, try to find the zeros of the function by setting each factor equal to zero and solving for $x$ .
When determining the multiplicity of each zero, count the number of times that factor appears in the factored form of the polynomial.

Final Thoughts

In conclusion, determining the zeros of a polynomial function and the multiplicity of each zero is an important concept in mathematics. By following the steps outlined in this article, you can find the zeros of a given polynomial function and list them from smallest to largest. Remember to always check your work and make sure that you have found all of the zeros of the function.

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Q: What is a polynomial function?

A: A polynomial function is a function of the form $G(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$ , where $a_n \neq 0$ and $n$ is a non-negative integer.

Q: What are the zeros of a polynomial function?

A: The zeros of a polynomial function are the values of $x$ for which $G(x) = 0$ . In other words, they are the solutions to the equation $G(x) = 0$ .

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

A: To determine the zeros of a polynomial function, you can try to factor the polynomial. If the polynomial can be factored, you can set each factor equal to zero and solve for $x$ . This will give you the zeros of the function.

Q: What is the multiplicity of a zero?

A: The multiplicity of a zero is the number of times that factor appears in the factored form of the polynomial. In other words, it is the number of times that the zero appears as a solution to the equation $G(x) = 0$ .

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

A: To determine the multiplicity of a zero, you can count the number of times that factor appears in the factored form of the polynomial.

Q: Can a polynomial function have complex zeros?

A: Yes, a polynomial function can have complex zeros. In fact, complex zeros are a common occurrence in polynomial functions.

Q: How do I determine the zeros of a polynomial function with complex zeros?

A: To determine the zeros of a polynomial function with complex zeros, you can try to factor the polynomial. If the polynomial can be factored, you can set each factor equal to zero and solve for $x$ . This will give you the zeros of the function.

Q: What is the difference between a real zero and a complex zero?

A: A real zero is a zero that is a real number, while a complex zero is a zero that is a complex number.

Q: Can a polynomial function have both real and complex zeros?

A: Yes, a polynomial function can have both real and complex zeros.

Q: How do I list the zeros of a polynomial function from smallest to largest?

A: To list the zeros of a polynomial function from smallest to largest, you can first determine the zeros of the function and then list them in order from smallest to largest.

Q: What is the importance of determining the zeros of a polynomial function?

A: Determining the zeros of a polynomial function is important because it allows you to understand the behavior of the function and make predictions about its behavior.

Q: Can you give an example of a polynomial function with real zeros?

A: Yes, an example of a polynomial function with real zeros is $G(x) = x^2 + 4x + 4$ . The zeros of this function are $x = -2$ and $x = -2$ , each with a multiplicity of 2.

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

A: Yes, an example of a polynomial function with complex zeros is $G(x) = x^3 + 3x^2 + 3x + 1$ . The zeros of this function are $x = -1$ , $x = -1 + i$ , and $x = -1 - i$ , each with a multiplicity of 1.

Q: What is the final answer to the problem of determining the zeros of a polynomial function?

A: The final answer to the problem of determining the zeros of a polynomial function is the list of zeros of the function, along with their multiplicities.

Q: Can you provide a summary of the main points of this article?

A: Yes, the main points of this article are:

A polynomial function is a function of the form $G(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$ , where $a_n \neq 0$ and $n$ is a non-negative integer.
The zeros of a polynomial function are the values of $x$ for which $G(x) = 0$ .
To determine the zeros of a polynomial function, you can try to factor the polynomial.
The multiplicity of a zero is the number of times that factor appears in the factored form of the polynomial.
A polynomial function can have complex zeros.
To determine the zeros of a polynomial function with complex zeros, you can try to factor the polynomial.
The zeros of a polynomial function can be listed from smallest to largest.

Q: What is the next step after determining the zeros of a polynomial function?

A: The next step after determining the zeros of a polynomial function is to use the zeros to understand the behavior of the function and make predictions about its behavior.