{ \begin{array}{l} f(x) = X^3 - 9x \\ g(x) = X^2 - 2x - 3 \end{array} \}$Which Of The Following Expressions Is Equivalent To ${$\frac{f(x)}{g(x)}\$}$ For ${$x \ \textgreater \ 3\$}$?A) ${$\frac{1}{x+1}\$}$ B)

Introduction

Rational functions are a fundamental concept in algebra, and simplifying them is a crucial skill for any math enthusiast. In this article, we will explore the process of simplifying rational functions, with a focus on the given expressions $f(x) = x^3 - 9x$ and $g(x) = x^2 - 2x - 3$. We will examine the expression $\frac{f(x)}{g(x)}$ and determine which of the given options is equivalent to it for $x > 3$.

Understanding Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In this case, we have the function $f(x) = x^3 - 9x$, which is a polynomial of degree 3, and the function $g(x) = x^2 - 2x - 3$, which is a polynomial of degree 2. The expression $\frac{f(x)}{g(x)}$ is a rational function, and our goal is to simplify it.

Simplifying Rational Functions

To simplify a rational function, we need to factor the numerator and denominator, and then cancel out any common factors. Let's start by factoring the numerator and denominator of the expression $\frac{f(x)}{g(x)}$.

Factoring the Numerator

The numerator of the expression is $f(x) = x^3 - 9x$. We can factor out an $x$ from the numerator, which gives us:

$$f(x) = x(x^2 - 9)$$

Factoring the Denominator

The denominator of the expression is $g(x) = x^2 - 2x - 3$. We can factor the denominator as follows:

$$g(x) = (x - 3)(x + 1)$$

Simplifying the Expression

Now that we have factored the numerator and denominator, we can simplify the expression by canceling out any common factors. In this case, we can cancel out the factor $(x - 3)$ from the numerator and denominator, which gives us:

$$\frac{f(x)}{g(x)} = \frac{x(x^2 - 9)}{(x - 3)(x + 1)} = \frac{x(x - 3)(x + 3)}{(x - 3)(x + 1)}$$

Canceling Out Common Factors

We can cancel out the factor $(x - 3)$ from the numerator and denominator, which gives us:

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

Simplifying the Expression Further

We can simplify the expression further by canceling out any common factors. In this case, we can cancel out the factor $x$ from the numerator and denominator, which gives us:

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

Evaluating the Expression

Now that we have simplified the expression, we can evaluate it for $x > 3$. We can substitute $x = 4$ into the expression, which gives us:

$$\frac{f(4)}{g(4)} = \frac{4 + 3}{1} = \frac{7}{1} = 7$$

Conclusion

In conclusion, the expression $\frac{f(x)}{g(x)}$ is equivalent to $\frac{x + 3}{1}$ for $x > 3$. This is the correct answer, and it can be verified by evaluating the expression for $x = 4$.

Discussion

The discussion category for this article is mathematics, and the topic is simplifying rational functions. The article provides a step-by-step guide on how to simplify rational functions, with a focus on the given expressions $f(x) = x^3 - 9x$ and $g(x) = x^2 - 2x - 3$. The article also provides a conclusion and a final answer.

Final Answer

The final answer is $\boxed{\frac{x + 3}{1}}$.

Options

The options for this problem are:

A) $\frac{1}{x+1}$ B) $\frac{x+3}{x-3}$ C) $\frac{x+3}{x+1}$ D) $\frac{x-3}{x+1}$

The correct answer is C) $\frac{x+3}{x+1}$.

Explanation

The explanation for this problem is that the expression $\frac{f(x)}{g(x)}$ can be simplified by canceling out common factors. The numerator and denominator can be factored, and then the common factor $(x - 3)$ can be canceled out. This gives us the simplified expression $\frac{x + 3}{x + 1}$. This expression is equivalent to the original expression for $x > 3$.

Key Concepts

The key concepts for this problem are:

• Rational functions
• Factoring polynomials
• Canceling out common factors
• Simplifying expressions

Applications

The applications of this problem are:

• Algebra
• Calculus
• Mathematics

Real-World Examples

The real-world examples of this problem are:

• Simplifying rational expressions in algebra
• Evaluating rational functions in calculus
• Solving problems in mathematics

Conclusion

In conclusion, the expression $\frac{f(x)}{g(x)}$ is equivalent to $\frac{x + 3}{x + 1}$ for $x > 3$. This is the correct answer, and it can be verified by evaluating the expression for $x = 4$. The discussion category for this article is mathematics, and the topic is simplifying rational functions. The article provides a step-by-step guide on how to simplify rational functions, with a focus on the given expressions $f(x) = x^3 - 9x$ and $g(x) = x^2 - 2x - 3$. The article also provides a conclusion and a final answer.

Introduction

In our previous article, we explored the process of simplifying rational functions, with a focus on the given expressions $f(x) = x^3 - 9x$ and $g(x) = x^2 - 2x - 3$. We determined that the expression $\frac{f(x)}{g(x)}$ is equivalent to $\frac{x + 3}{x + 1}$ for $x > 3$. In this article, we will provide a Q&A guide to help you better understand the process of simplifying rational functions.

Q: What is a rational function?

A: A rational function is a function that can be expressed as the ratio of two polynomials. In this case, we have the function $f(x) = x^3 - 9x$, which is a polynomial of degree 3, and the function $g(x) = x^2 - 2x - 3$, which is a polynomial of degree 2.

Q: How do I simplify a rational function?

A: To simplify a rational function, you need to factor the numerator and denominator, and then cancel out any common factors. Let's start by factoring the numerator and denominator of the expression $\frac{f(x)}{g(x)}$.

Q: What is the first step in simplifying a rational function?

A: The first step in simplifying a rational function is to factor the numerator and denominator. In this case, we can factor the numerator as $f(x) = x(x^2 - 9)$ and the denominator as $g(x) = (x - 3)(x + 1)$.

Q: How do I cancel out common factors?

A: To cancel out common factors, you need to identify the common factors in the numerator and denominator, and then cancel them out. In this case, we can cancel out the factor $(x - 3)$ from the numerator and denominator.

Q: What is the simplified expression?

A: The simplified expression is $\frac{x + 3}{x + 1}$.

Q: Is the simplified expression equivalent to the original expression for $x > 3$?

A: Yes, the simplified expression is equivalent to the original expression for $x > 3$.

Q: How do I evaluate the simplified expression?

A: To evaluate the simplified expression, you can substitute a value of $x$ into the expression. In this case, we can substitute $x = 4$ into the expression, which gives us $\frac{4 + 3}{4 + 1} = \frac{7}{5}$.

Q: What is the final answer?

A: The final answer is $\boxed{\frac{x + 3}{x + 1}}$.

Q: What are some common mistakes to avoid when simplifying rational functions?

A: Some common mistakes to avoid when simplifying rational functions include:

• Not factoring the numerator and denominator
• Not canceling out common factors
• Not evaluating the simplified expression

Q: How do I apply the process of simplifying rational functions to real-world problems?

A: To apply the process of simplifying rational functions to real-world problems, you need to identify the rational function, factor the numerator and denominator, cancel out common factors, and then evaluate the simplified expression.

Q: What are some real-world examples of simplifying rational functions?

A: Some real-world examples of simplifying rational functions include:

• Simplifying rational expressions in algebra
• Evaluating rational functions in calculus
• Solving problems in mathematics

Conclusion

In conclusion, simplifying rational functions is a crucial skill for any math enthusiast. By following the steps outlined in this article, you can simplify rational functions and apply the process to real-world problems. Remember to factor the numerator and denominator, cancel out common factors, and then evaluate the simplified expression. With practice and patience, you can become proficient in simplifying rational functions and apply the process to a wide range of problems.

Final Answer

The final answer is $\boxed{\frac{x + 3}{x + 1}}$.

Options

The options for this problem are:

A) $\frac{1}{x+1}$ B) $\frac{x+3}{x-3}$ C) $\frac{x+3}{x+1}$ D) $\frac{x-3}{x+1}$

The correct answer is C) $\frac{x+3}{x+1}$.

Explanation

The explanation for this problem is that the expression $\frac{f(x)}{g(x)}$ can be simplified by canceling out common factors. The numerator and denominator can be factored, and then the common factor $(x - 3)$ can be canceled out. This gives us the simplified expression $\frac{x + 3}{x + 1}$. This expression is equivalent to the original expression for $x > 3$.

Key Concepts

The key concepts for this problem are:

• Rational functions
• Factoring polynomials
• Canceling out common factors
• Simplifying expressions

Applications

The applications of this problem are:

• Algebra
• Calculus
• Mathematics

Real-World Examples

The real-world examples of this problem are:

• Simplifying rational expressions in algebra
• Evaluating rational functions in calculus
• Solving problems in mathematics

Conclusion

In conclusion, simplifying rational functions is a crucial skill for any math enthusiast. By following the steps outlined in this article, you can simplify rational functions and apply the process to real-world problems. Remember to factor the numerator and denominator, cancel out common factors, and then evaluate the simplified expression. With practice and patience, you can become proficient in simplifying rational functions and apply the process to a wide range of problems.