Which Of The Following Describes The Zeroes Of The Graph Of $f(x)=3x^6+30x^5+75x^4$?A. -5 With Multiplicity 2 And $\frac{1}{3}$ With Multiplicity 4 B. 5 With Multiplicity 2 And $\frac{1}{3}$ With Multiplicity 4 C. -5 With

Introduction to Polynomial Functions

Polynomial functions are a fundamental concept in mathematics, and understanding their properties is crucial for solving various mathematical problems. 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. In this article, we will focus on the zeroes of a polynomial function, specifically the function $f(x)=3x^6+30x^5+75x^4$.

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 $f(x)$ is a polynomial function, then the zeroes of $f(x)$ are the values of $x$ such that $f(x) = 0$. The zeroes of a polynomial function are also known as the roots of the function.

Types of Zeroes

There are two types of zeroes of a polynomial function: real zeroes and complex zeroes. Real zeroes are the values of the variable that make the function equal to zero, and they can be either positive or negative. Complex zeroes, on the other hand, are the values of the variable that make the function equal to zero, but they are not real numbers.

Multiplicity of Zeroes

The multiplicity of a zero is the number of times the zero appears in the factorization of the polynomial function. In other words, if a zero has a multiplicity of $n$, then the zero appears $n$ times in the factorization of the polynomial function.

Factoring the Polynomial Function

To find the zeroes of the polynomial function $f(x)=3x^6+30x^5+75x^4$, we need to factor the function. Factoring a polynomial function involves expressing the function as a product of simpler polynomials. In this case, we can factor the function as follows:

$f(x) = 3x^4(x^2+10x+25)$

Solving for the Zeroes

To find the zeroes of the function, we need to solve for the values of $x$ that make the function equal to zero. We can do this by setting the function equal to zero and solving for $x$.

$3x^4(x^2+10x+25) = 0$

Finding the Zeroes

We can find the zeroes of the function by setting each factor equal to zero and solving for $x$.

$3x^4 = 0 \Rightarrow x = 0$

$x^2+10x+25 = 0 \Rightarrow (x+5)^2 = 0 \Rightarrow x = -5$

Multiplicity of the Zeroes

We can find the multiplicity of each zero by looking at the factorization of the polynomial function.

$3x^4(x^2+10x+25) = 0$

The zero $x = 0$ has a multiplicity of 4, and the zero $x = -5$ has a multiplicity of 2.

Conclusion

In conclusion, the zeroes of the polynomial function $f(x)=3x^6+30x^5+75x^4$ are $x = 0$ with multiplicity 4 and $x = -5$ with multiplicity 2.

Answer

The correct answer is A. -5 with multiplicity 2 and $\frac{1}{3}$ with multiplicity 4 is incorrect, the correct answer is 0 with multiplicity 4 and -5 with multiplicity 2.

Final Answer

The final answer is 0 with multiplicity 4 and -5 with multiplicity 2.

Introduction

In our previous article, we discussed the zeroes of a polynomial function, specifically the function $f(x)=3x^6+30x^5+75x^4$. We found that the zeroes of the function are $x = 0$ with multiplicity 4 and $x = -5$ with multiplicity 2. In this article, we will answer some frequently asked questions about the zeroes of a polynomial function.

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

A: The terms "zero" and "root" are often used interchangeably to refer to the values of the variable that make the function equal to zero. However, some mathematicians make a distinction between the two terms. 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 it is also a solution to the equation.

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

A: To find the zeroes of a polynomial function, you need to factor the function and set each factor equal to zero. You can then solve for the values of the variable 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 zero appears in the factorization of the polynomial function. In other words, if a zero has a multiplicity of $n$, then the zero appears $n$ times in the factorization of the polynomial function.

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

A: To determine the multiplicity of a zero, you need to look at the factorization of the polynomial function. If a zero appears $n$ times in the factorization, then the zero has a multiplicity of $n$.

Q: Can a zero have a negative multiplicity?

A: No, a zero cannot have a negative multiplicity. The multiplicity of a zero is always a non-negative integer.

Q: Can a polynomial function have more than one zero with the same multiplicity?

A: Yes, a polynomial function can have more than one zero with the same multiplicity. For example, the polynomial function $f(x) = x^2(x-1)^2$ has two zeroes, $x = 0$ and $x = 1$, both with multiplicity 2.

Q: How do I graph a polynomial function with multiple zeroes?

A: To graph a polynomial function with multiple zeroes, you need to plot the zeroes on the graph and then use the multiplicity of each zero to determine the behavior of the graph near each zero.

Q: Can a polynomial function have a zero that is not a real number?

A: Yes, a polynomial function can have a zero that is not a real number. For example, the polynomial function $f(x) = x^2 + 1$ has a zero at $x = i$, where $i$ is the imaginary unit.

Conclusion

In conclusion, the zeroes of a polynomial function are the values of the variable that make the function equal to zero. The multiplicity of a zero is the number of times the zero appears in the factorization of the polynomial function. We hope that this Q&A article has helped to clarify any questions you may have had about the zeroes of a polynomial function.

Frequently Asked Questions

• Q: What is the difference between a zero and a root of a polynomial function?
• A: The terms "zero" and "root" are often used interchangeably to refer to the values of the variable that make the function equal to zero.
• Q: How do I find the zeroes of a polynomial function?
• A: To find the zeroes of a polynomial function, you need to factor the function and set each factor equal to zero.
• Q: What is the multiplicity of a zero?
• A: The multiplicity of a zero is the number of times the zero appears in the factorization of the polynomial function.

Additional Resources

• For more information on the zeroes of a polynomial function, please see our previous article on the topic.
• For more information on graphing polynomial functions, please see our article on graphing polynomial functions.
• For more information on the properties of polynomial functions, please see our article on the properties of polynomial functions.