Polynomial And Rational Functions Test Part 1Which Of The Following Is An Eighth-degree Polynomial Function? Select All That Apply.(1 Point)A. F ( X ) = ( X 4 + X ) 2 F(x) = \left(x^4 + X\right)^2 F ( X ) = ( X 4 + X ) 2 B. F ( X ) = 1 X 8 − 64 F(x) = \frac{1}{x^8 - 64} F ( X ) = X 8 − 64 1 ​ C. $f(x) = 10x -

Introduction

Polynomial and rational functions are fundamental concepts in algebra and mathematics. They are used to model various real-world phenomena, such as population growth, chemical reactions, and electrical circuits. In this test, we will focus on identifying and analyzing polynomial and rational functions. Specifically, we will determine which of the given functions is an eighth-degree polynomial function.

What is a Polynomial Function?

A polynomial function is a function that can be written in the form:

$$f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$$

where $a_n \neq 0$ and $n$ is a non-negative integer. The degree of a polynomial function is the highest power of $x$ in the function.

What is a Rational Function?

A rational function is a function that can be written in the form:

$$f(x) = \frac{p(x)}{q(x)}$$

where $p(x)$ and $q(x)$ are polynomial functions and $q(x) \neq 0$. The degree of a rational function is the difference between the degrees of the numerator and denominator.

Eighth-Degree Polynomial Function

An eighth-degree polynomial function is a polynomial function of degree 8. In other words, the highest power of $x$ in the function is 8.

Analyzing the Options

Let's analyze the given options:

Option A: $f(x) = \left(x^4 + x\right)^2$

To determine if this function is an eighth-degree polynomial function, we need to expand the expression:

$$f(x) = \left(x^4 + x\right)^2 = x^8 + 2x^5 + x^2$$

The highest power of $x$ in this function is 8, so it is indeed an eighth-degree polynomial function.

Option B: $f(x) = \frac{1}{x^8 - 64}$

To determine if this function is an eighth-degree polynomial function, we need to analyze the denominator:

$$x^8 - 64 = (x^4 - 8)(x^4 + 8)$$

The degree of the denominator is 8, but the degree of the numerator is 0. Therefore, the degree of the rational function is -8, which is not a valid degree. However, we can rewrite the function as:

$$f(x) = \frac{1}{(x^4 - 8)(x^4 + 8)} = \frac{1}{x^8 - 64}$$

This function is not an eighth-degree polynomial function, but rather a rational function with a degree of -8.

Option C: $f(x) = 10x^8$

This function is clearly an eighth-degree polynomial function, as the highest power of $x$ is 8.

Conclusion

In conclusion, the options that are eighth-degree polynomial functions are:

• Option A: $f(x) = \left(x^4 + x\right)^2$
• Option C: $f(x) = 10x^8$

These functions have the highest power of $x$ equal to 8, making them eighth-degree polynomial functions.

Practice Problems

1. Which of the following is a sixth-degree polynomial function?
   • A. $f(x) = x^6 + 2x^3 + 1$
   • B. $f(x) = \frac{1}{x^6 - 1}$
   • C. $f(x) = 3x^6 - 2x^3 + 1$
2. Which of the following is a rational function with a degree of 3?
   • A. $f(x) = \frac{x^3 + 1}{x^2 + 1}$
   • B. $f(x) = \frac{x^2 + 1}{x^3 + 1}$
   • C. $f(x) = \frac{x^3 + 1}{x^3 - 1}$

Answer Key

1. A and C
2. A and C
   Polynomial and Rational Functions Test Part 2: Q&A =====================================================

Introduction

In the previous article, we discussed polynomial and rational functions, and identified which of the given functions is an eighth-degree polynomial function. In this article, we will provide a Q&A section to help you better understand the concepts and prepare for your test.

Q&A

Q: What is the difference between a polynomial function and a rational function?

A: A polynomial function is a function that can be written in the form:

$$f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$$

where $a_n \neq 0$ and $n$ is a non-negative integer.

A rational function, on the other hand, is a function that can be written in the form:

$$f(x) = \frac{p(x)}{q(x)}$$

where $p(x)$ and $q(x)$ are polynomial functions and $q(x) \neq 0$.

Q: What is the degree of a polynomial function?

A: The degree of a polynomial function is the highest power of $x$ in the function.

Q: What is the degree of a rational function?

A: The degree of a rational function is the difference between the degrees of the numerator and denominator.

Q: How do I determine if a function is a polynomial function or a rational function?

A: To determine if a function is a polynomial function or a rational function, you need to analyze the function and determine if it can be written in the form of a polynomial function or a rational function.

Q: What is an eighth-degree polynomial function?

A: An eighth-degree polynomial function is a polynomial function of degree 8. In other words, the highest power of $x$ in the function is 8.

Q: How do I determine if a function is an eighth-degree polynomial function?

A: To determine if a function is an eighth-degree polynomial function, you need to analyze the function and determine if the highest power of $x$ is 8.

Q: What are some examples of polynomial functions?

A: Some examples of polynomial functions include:

• $f(x) = x^2 + 2x + 1$
• $f(x) = 3x^4 - 2x^2 + 1$
• $f(x) = x^6 + 2x^3 + 1$

Q: What are some examples of rational functions?

A: Some examples of rational functions include:

• $f(x) = \frac{x^2 + 1}{x + 1}$
• $f(x) = \frac{x^3 + 1}{x^2 + 1}$
• $f(x) = \frac{x^4 + 1}{x^2 - 1}$

Practice Problems

1. Determine if the following function is a polynomial function or a rational function:

$$f(x) = \frac{x^2 + 1}{x + 1}$$

2. Determine if the following function is an eighth-degree polynomial function:

$$f(x) = x^8 + 2x^5 + x^2$$

3. Determine if the following function is a polynomial function or a rational function:

$$f(x) = \frac{x^3 + 1}{x^2 + 1}$$

Answer Key

1. Rational function
2. Eighth-degree polynomial function
3. Rational function

Conclusion

In conclusion, we hope this Q&A section has helped you better understand the concepts of polynomial and rational functions. Remember to practice and review the concepts to prepare for your test. Good luck!