Which Of The Following Expressions Is Equivalent To F ( X ) G ( X ) \frac{f(x)}{g(x)} G ( X ) F ( X ) ​ , For X \textgreater 3 X\ \textgreater \ 3 X \textgreater 3 ?A) 1 X + 1 \frac{1}{x+1} X + 1 1 ​ B) X + 3 X + 1 \frac{x+3}{x+1} X + 1 X + 3 ​ C) X ( X − 3 ) X + 1 \frac{x(x-3)}{x+1} X + 1 X ( X − 3 ) ​ D) X ( X + 3 ) X + 1 \frac{x(x+3)}{x+1} X + 1 X ( X + 3 ) ​

Introduction

Rational expressions 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 expressions, with a focus on identifying equivalent expressions. We will examine a specific problem, where we need to determine which of the given expressions is equivalent to $\frac{f(x)}{g(x)}$ for $x > 3$.

Understanding Rational Expressions

A rational expression is a fraction in which the numerator and denominator are both polynomials. Rational expressions can be simplified by canceling out common factors between the numerator and denominator. This process is known as factoring.

Factoring Rational Expressions

To factor a rational expression, we need to identify the greatest common factor (GCF) of the numerator and denominator. The GCF is the largest expression that divides both the numerator and denominator without leaving a remainder.

For example, consider the rational expression $\frac{6x^2}{12x}$. To simplify this expression, we need to find the GCF of the numerator and denominator. The GCF of $6x^2$ and $12x$ is $6x$. We can cancel out the GCF from both the numerator and denominator to get $\frac{x}{2}$.

Simplifying Rational Expressions with Variables

When simplifying rational expressions with variables, we need to be careful not to cancel out any common factors that may be hidden in the variables. For example, consider the rational expression $\frac{x^2+3x}{x+1}$. To simplify this expression, we need to factor the numerator and denominator. The numerator can be factored as $(x+1)(x+3)$, and the denominator is already factored as $(x+1)$.

We can cancel out the common factor $(x+1)$ from both the numerator and denominator to get $\frac{x+3}{1}$, which simplifies to $x+3$.

The Problem: Which Expression is Equivalent to $\frac{f(x)}{g(x)}$?

Now that we have a good understanding of simplifying rational expressions, let's tackle the problem at hand. We are given four expressions, and we need to determine which one is equivalent to $\frac{f(x)}{g(x)}$ for $x > 3$.

The four expressions are:

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

To determine which expression is equivalent to $\frac{f(x)}{g(x)}$, we need to analyze each expression and see if it can be simplified to the form $\frac{f(x)}{g(x)}$.

Analyzing Expression A

Expression A is $\frac{1}{x+1}$. This expression cannot be simplified further, as there are no common factors between the numerator and denominator.

Analyzing Expression B

Expression B is $\frac{x+3}{x+1}$. This expression can be simplified by canceling out the common factor $(x+1)$ from both the numerator and denominator. This leaves us with $x+3$.

Analyzing Expression C

Expression C is $\frac{x(x-3)}{x+1}$. This expression can be simplified by canceling out the common factor $(x+1)$ from both the numerator and denominator. This leaves us with $x(x-3)$.

Analyzing Expression D

Expression D is $\frac{x(x+3)}{x+1}$. This expression can be simplified by canceling out the common factor $(x+1)$ from both the numerator and denominator. This leaves us with $x(x+3)$.

Conclusion

Based on our analysis, we can see that Expression B is the only one that can be simplified to the form $\frac{f(x)}{g(x)}$. The other expressions either cannot be simplified further or result in a different form.

Therefore, the correct answer is:

B) $\frac{x+3}{x+1}$

This expression is equivalent to $\frac{f(x)}{g(x)}$ for $x > 3$.

Final Thoughts

Simplifying rational expressions is an essential skill for any math enthusiast. By understanding the process of factoring and canceling out common factors, we can simplify complex rational expressions and identify equivalent expressions. In this article, we explored a specific problem and analyzed each expression to determine which one is equivalent to $\frac{f(x)}{g(x)}$ for $x > 3$. We hope this article has provided valuable insights and helped you develop your skills in simplifying rational expressions.

Introduction

In our previous article, we explored the process of simplifying rational expressions and identified the equivalent expression to $\frac{f(x)}{g(x)}$ for $x > 3$. In this article, we will provide a Q&A guide to help you better understand the concepts and techniques involved in simplifying rational expressions.

Q1: What is a rational expression?

A rational expression is a fraction in which the numerator and denominator are both polynomials. Rational expressions can be simplified by canceling out common factors between the numerator and denominator.

Q2: How do I simplify a rational expression?

To simplify a rational expression, you need to identify the greatest common factor (GCF) of the numerator and denominator. The GCF is the largest expression that divides both the numerator and denominator without leaving a remainder. You can then cancel out the GCF from both the numerator and denominator to simplify the expression.

Q3: What is factoring?

Factoring is the process of expressing a polynomial as a product of simpler polynomials. Factoring is an essential technique in simplifying rational expressions, as it allows you to identify common factors between the numerator and denominator.

Q4: How do I factor a rational expression?

To factor a rational expression, you need to identify the GCF of the numerator and denominator. You can then factor the numerator and denominator separately, and cancel out any common factors.

Q5: What is the difference between a rational expression and a polynomial?

A rational expression is a fraction in which the numerator and denominator are both polynomials. A polynomial, on the other hand, is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.

Q6: Can I simplify a rational expression with variables?

Yes, you can simplify a rational expression with variables. However, you need to be careful not to cancel out any common factors that may be hidden in the variables.

Q7: How do I simplify a rational expression with variables?

To simplify a rational expression with variables, you need to factor the numerator and denominator separately, and cancel out any common factors.

Q8: What is the equivalent expression to $\frac{f(x)}{g(x)}$ for $x > 3$?

The equivalent expression to $\frac{f(x)}{g(x)}$ for $x > 3$ is $\frac{x+3}{x+1}$.

Q9: Can I simplify a rational expression with a negative exponent?

Yes, you can simplify a rational expression with a negative exponent. However, you need to be careful not to cancel out any common factors that may be hidden in the negative exponent.

Q10: How do I simplify a rational expression with a negative exponent?

To simplify a rational expression with a negative exponent, you need to rewrite the expression with a positive exponent, and then simplify it using the techniques described above.

Conclusion

Simplifying rational expressions is an essential skill for any math enthusiast. By understanding the concepts and techniques involved in simplifying rational expressions, you can simplify complex expressions and identify equivalent expressions. We hope this Q&A guide has provided valuable insights and helped you develop your skills in simplifying rational expressions.

Final Thoughts

Simplifying rational expressions is a crucial skill in mathematics, and it has numerous applications in various fields, including science, engineering, and economics. By mastering the techniques involved in simplifying rational expressions, you can solve complex problems and make informed decisions. We hope this article has provided a comprehensive guide to simplifying rational expressions and has helped you develop your skills in this area.

Additional Resources

If you want to learn more about simplifying rational expressions, we recommend the following resources:

• Khan Academy: Simplifying Rational Expressions
• Mathway: Simplifying Rational Expressions
• Wolfram Alpha: Simplifying Rational Expressions

These resources provide a comprehensive guide to simplifying rational expressions, including examples, exercises, and interactive tools.