Listing the Zeros According to Their Multiplicities

Oct 13, 2025 by windailys.com 40 views

In this article, we will delve into the world of polynomial functions and explore the concept of zeros and their multiplicities. We will examine a specific polynomial function, $f(x) = 5(x+11)(x+3)(x-11)^2(x-4)$ , and list each of its zeros according to their multiplicities.

What are Zeros and Multiplicities?

In mathematics, 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. The multiplicity of a zero is the number of times the factor $(x - a)$ appears in the polynomial function.

The Polynomial Function $f(x)$

The given polynomial function is $f(x) = 5(x+11)(x+3)(x-11)^2(x-4)$ . To find the zeros of this function, we need to set $f(x) = 0$ and solve for $x$ .

Finding the Zeros of $f(x)$

To find the zeros of $f(x)$ , we can set each factor equal to zero and solve for $x$ . This will give us the values of $x$ that make the function equal to zero.

Zero 1: $x + 11 = 0$

Solving for $x$ , we get:

$x + 11 = 0$ $x = -11$

Zero 2: $x + 3 = 0$

Solving for $x$ , we get:

$x + 3 = 0$ $x = -3$

Zero 3: $(x - 11)^2 = 0$

Solving for $x$ , we get:

$(x - 11)^2 = 0$ $x - 11 = 0$ $x = 11$

Since the factor $(x - 11)$ appears twice in the polynomial function, the multiplicity of this zero is 2.

Zero 4: $x - 4 = 0$

Solving for $x$ , we get:

$x - 4 = 0$ $x = 4$

Listing the Zeros According to Their Multiplicities

Now that we have found the zeros of the polynomial function, we can list them according to their multiplicities.

Multiplicity 1: $x = -11$

Zero: $x = -11$
Multiplicity: 1

Multiplicity 1: $x = -3$

Zero: $x = -3$
Multiplicity: 1

Multiplicity 2: $x = 11$

Zero: $x = 11$
Multiplicity: 2

Multiplicity 1: $x = 4$

Zero: $x = 4$
Multiplicity: 1

Conclusion

In this article, we have explored the concept of zeros and multiplicities of a polynomial function. We have examined a specific polynomial function, $f(x) = 5(x+11)(x+3)(x-11)^2(x-4)$ , and listed each of its zeros according to their multiplicities. We have found that the zeros of the function are $x = -11$ , $x = -3$ , $x = 11$ , and $x = 4$ , with multiplicities of 1, 1, 2, and 1, respectively.

Discussion Category: Mathematics

This article has provided a comprehensive overview of the concept of zeros and multiplicities of a polynomial function. The discussion category of mathematics is relevant to this article as it deals with the mathematical concepts of polynomial functions, zeros, and multiplicities.

Key Takeaways

A zero of a polynomial function is a value of $x$ that makes the function equal to zero.
The multiplicity of a zero is the number of times the factor $(x - a)$ appears in the polynomial function.
To find the zeros of a polynomial function, we need to set each factor equal to zero and solve for $x$ .
The zeros of a polynomial function can be listed according to their multiplicities.

References

[1] "Polynomial Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/polynomial.html
[2] "Zeros and Multiplicities" by Khan Academy. Retrieved from <https://www.khanacademy.org/math/algebra/x2f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d7f0f7d
Q&A: Zeros and Multiplicities of Polynomial Functions =====================================================

In our previous article, we explored the concept of zeros and multiplicities of polynomial functions. We examined a specific polynomial function, $f(x) = 5(x+11)(x+3)(x-11)^2(x-4)$ , and listed each of its zeros according to their multiplicities. In this article, we will answer some frequently asked questions about zeros and multiplicities of polynomial functions.

Q: What is a zero of a polynomial function?

A: 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.

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

A: To find the zeros of a polynomial function, you need to set each factor equal to zero and solve for $x$ . This will give you 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 $(x - a)$ appears in the polynomial function. For example, if the factor $(x - 2)$ appears twice in the polynomial function, then the multiplicity of the zero $x = 2$ is 2.

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

A: To determine the multiplicity of a zero, you need to examine the polynomial function and count the number of times the factor $(x - a)$ appears. You can also use the fact that the multiplicity of a zero is equal to the exponent of the factor $(x - a)$ in the polynomial function.

Q: Can a zero have a multiplicity of zero?

A: No, a zero cannot have a multiplicity of zero. This is because the multiplicity of a zero is defined as the number of times the factor $(x - a)$ appears in the polynomial function, and if the factor $(x - a)$ does not appear at all, then the multiplicity is not defined.

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: Can a polynomial function have a zero with a negative multiplicity?

A: No, a polynomial function cannot have a zero with a negative multiplicity. This is because the multiplicity of a zero is defined as the number of times the factor $(x - a)$ appears in the polynomial function, and if the factor $(x - a)$ appears a negative number of times, then the multiplicity is not defined.

Q: Can a polynomial function have a zero with a fractional multiplicity?

A: No, a polynomial function cannot have a zero with a fractional multiplicity. This is because the multiplicity of a zero is defined as the number of times the factor $(x - a)$ appears in the polynomial function, and if the factor $(x - a)$ appears a fractional number of times, then the multiplicity is not defined.

Conclusion

In this article, we have answered some frequently asked questions about zeros and multiplicities of polynomial functions. We have explored the concept of zeros and multiplicities, and provided examples to illustrate the concepts. We hope that this article has been helpful in clarifying the concepts of zeros and multiplicities of polynomial functions.