Find The Zeros And Give The Multiplicity Of Each. List The Zeros In Order From Least To Greatest.Given: $f(x)=x^3(x+5)^4(x-3)^2$1. Zero At $\square$ With Multiplicity $\square$ 2. Zero At $\square$ With Multiplicity

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Introduction

In algebra, 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 zeros of a polynomial function are the values of the variable that make the function equal to zero. In this article, we will discuss how to find the zeros and multiplicity of each zero of a given polynomial function.

Understanding the Concept of Multiplicity

The multiplicity of a zero is the number of times that the factor corresponding to that zero appears in the factored form of the polynomial. For example, if a polynomial function has a factor of (x−a)2(x - a)^2, then the zero aa has a multiplicity of 2.

Given Polynomial Function

The given polynomial function is f(x)=x3(x+5)4(x−3)2f(x) = x^3(x + 5)^4(x - 3)^2. To find the zeros and multiplicity of each zero, we need to factorize the polynomial function.

Factoring the Polynomial Function

We can factorize the polynomial function as follows:

f(x)=x3(x+5)4(x−3)2f(x) = x^3(x + 5)^4(x - 3)^2

=x3(x+5)4(x−3)2= x^3(x + 5)^4(x - 3)^2

=(x)(x)(x)(x+5)(x+5)(x+5)(x+5)(x+5)(x−3)(x−3)= (x)(x)(x)(x + 5)(x + 5)(x + 5)(x + 5)(x + 5)(x - 3)(x - 3)

Finding the Zeros

From the factored form of the polynomial function, we can see that the zeros are x=0x = 0, x=−5x = -5, and x=3x = 3. We can now find the multiplicity of each zero.

Multiplicity of Each Zero

  • Zero at x=0x = 0 with multiplicity 3: This is because the factor (x)(x) appears three times in the factored form of the polynomial function.
  • Zero at x=−5x = -5 with multiplicity 4: This is because the factor (x+5)(x + 5) appears four times in the factored form of the polynomial function.
  • Zero at x=3x = 3 with multiplicity 2: This is because the factor (x−3)(x - 3) appears two times in the factored form of the polynomial function.

Listing the Zeros in Order from Least to Greatest

The zeros of the polynomial function in order from least to greatest are:

  • x=−5x = -5
  • x=0x = 0
  • x=3x = 3

Conclusion

In this article, we discussed how to find the zeros and multiplicity of each zero of a given polynomial function. We used the factored form of the polynomial function to find the zeros and multiplicity of each zero. We also listed the zeros in order from least to greatest. The zeros of the polynomial function are x=−5x = -5, x=0x = 0, and x=3x = 3, with multiplicities 4, 3, and 2, respectively.

Frequently Asked Questions

Q: What is the multiplicity of a zero in a polynomial function?

A: The multiplicity of a zero is the number of times that the factor corresponding to that zero appears in the factored form of the polynomial.

Q: How do I find the zeros and multiplicity of each zero of a polynomial function?

A: To find the zeros and multiplicity of each zero of a polynomial function, you need to factorize the polynomial function and then find the factors that correspond to each zero.

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, but it can also be a complex number.

Q: How do I list the zeros of a polynomial function in order from least to greatest?

A: To list the zeros of a polynomial function in order from least to greatest, you need to first find the zeros and then arrange them in order from least to greatest.

References

Introduction

In our previous article, we discussed how to find the zeros and multiplicity of each zero of a given polynomial function. In this article, we will answer some frequently asked questions related to finding zeros and multiplicity of a polynomial function.

Q&A

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, but it can also be a complex number.

Q: How do I find the zeros and multiplicity of each zero of a polynomial function?

A: To find the zeros and multiplicity of each zero of a polynomial function, you need to factorize the polynomial function and then find the factors that correspond to each zero.

Q: What is the multiplicity of a zero in a polynomial function?

A: The multiplicity of a zero is the number of times that the factor corresponding to that zero appears in the factored form of the polynomial.

Q: How do I list the zeros of a polynomial function in order from least to greatest?

A: To list the zeros of a polynomial function in order from least to greatest, you need to first find the zeros and then arrange them in order from least to greatest.

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.

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

A: To find the zeros of a polynomial function with complex coefficients, you need to use the quadratic formula or other methods to find the complex roots of the polynomial.

Q: Can a polynomial function have a zero with a multiplicity greater than the degree of the polynomial?

A: No, a polynomial function cannot have a zero with a multiplicity greater than the degree of the polynomial. The multiplicity of a zero is always less than or equal to the degree of the polynomial.

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

A: To find the zeros of a polynomial function with repeated factors, you need to factorize the polynomial function and then find the factors that correspond to each zero.

Q: Can a polynomial function have a zero with a multiplicity of 0?

A: No, a polynomial function cannot have a zero with a multiplicity of 0. The multiplicity of a zero is always a positive integer.

Conclusion

In this article, we answered some frequently asked questions related to finding zeros and multiplicity of a polynomial function. We hope that this article has been helpful in clarifying some of the concepts related to finding zeros and multiplicity of a polynomial function.

Frequently Asked Questions

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, but it can also be a complex number.

Q: How do I find the zeros and multiplicity of each zero of a polynomial function?

A: To find the zeros and multiplicity of each zero of a polynomial function, you need to factorize the polynomial function and then find the factors that correspond to each zero.

Q: What is the multiplicity of a zero in a polynomial function?

A: The multiplicity of a zero is the number of times that the factor corresponding to that zero appears in the factored form of the polynomial.

Q: How do I list the zeros of a polynomial function in order from least to greatest?

A: To list the zeros of a polynomial function in order from least to greatest, you need to first find the zeros and then arrange them in order from least to greatest.

References