Simplifying Rational Expressions A Step-by-Step Guide

Jul 10, 2025 by ADMIN 54 views

Question 4 of 25: Simplifying Rational Expressions

In this article, we will delve into the process of simplifying rational expressions, focusing on the specific question: Which of the following is equal to the rational expression $\frac{x^2-9}{(x-1)(x-3)}$ when $x \neq 1$ or 3? This question tests our understanding of factoring, simplifying, and recognizing equivalent expressions. We will break down the steps involved in simplifying the given rational expression and identify the correct answer choice. By the end of this guide, you will have a solid understanding of how to tackle similar problems involving rational expressions.

Understanding Rational Expressions

Before we dive into the problem, let's first define what a rational expression is. A rational expression is essentially a fraction where the numerator and denominator are polynomials. Polynomials are algebraic expressions that consist of variables and coefficients, combined using addition, subtraction, and non-negative integer exponents. Examples of polynomials include $x^2 - 9$ , $(x - 1)$ , and $(x - 3)$ .

Rational expressions often need to be simplified to make them easier to work with. Simplification involves reducing the expression to its lowest terms by canceling out common factors in the numerator and denominator. This process is similar to simplifying numerical fractions, such as reducing $\frac{6}{8}$ to $\frac{3}{4}$ .

The key to simplifying rational expressions lies in factoring. Factoring a polynomial means expressing it as a product of two or more simpler polynomials. For instance, the polynomial $x^2 - 9$ can be factored into $(x + 3)(x - 3)$ . Factoring allows us to identify common factors between the numerator and denominator, which can then be canceled out.

The Importance of Restrictions

Notice the condition $x \neq 1$ or 3 in the question. This condition is crucial because it highlights the concept of restrictions on the variable $x$ . Rational expressions are undefined when the denominator is equal to zero. In our case, the denominator is $(x - 1)(x - 3)$ . If $x = 1$ or $x = 3$ , the denominator becomes zero, making the expression undefined. Therefore, we must exclude these values from the domain of the expression.

Restrictions are an essential part of working with rational expressions. Always identify the values of the variable that make the denominator zero and exclude them from the solution set. These restrictions ensure that the expression remains mathematically valid.

Step-by-Step Simplification

Now, let's simplify the given rational expression step by step:

1. Factor the Numerator: The numerator is $x^2 - 9$ . This is a difference of squares, which can be factored using the formula $a^2 - b^2 = (a + b)(a - b)$ . In our case, $a = x$ and $b = 3$ , so we have:

$x^2 - 9 = (x + 3)(x - 3)$

2. Rewrite the Expression with Factored Numerator: Substitute the factored numerator back into the rational expression:

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

3. Cancel Common Factors: Observe that both the numerator and denominator have a common factor of $(x - 3)$ . We can cancel this factor out, provided that $x \neq 3$ (which is already given as a restriction):

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

4. Simplified Expression: The simplified rational expression is $\frac{x + 3}{x - 1}$ . This expression is equivalent to the original expression, except for the restriction that $x \neq 1$ and $x \neq 3$ .

Identifying the Correct Answer

Now that we have simplified the expression, we can compare it to the answer choices provided:

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

Comparing our simplified expression $\frac{x + 3}{x - 1}$ to the options, we can see that option A matches perfectly. Therefore, the correct answer is A.

Analyzing the Incorrect Options

To further solidify our understanding, let's analyze why the other options are incorrect:

Option B: $\frac{x - 1}{x - 3}$ This option is the reciprocal of a factor in the original expression but does not represent the fully simplified form. It fails to account for the factoring of the numerator and the cancellation of common factors.
Option C: $\frac{x - 3}{x - 1}$ This option has the correct factors in the numerator and denominator but has the subtraction sign in the wrong place. It does not match the result of correctly factoring and simplifying the original expression.

By understanding why these options are incorrect, we reinforce our grasp of the simplification process and the importance of accurate factoring and cancellation.

Tips for Simplifying Rational Expressions

Here are some helpful tips to keep in mind when simplifying rational expressions:

Factor Completely: Always factor both the numerator and the denominator completely. This allows you to identify all common factors that can be canceled out.
Identify Restrictions: Determine the values of the variable that make the denominator zero and exclude them from the solution set. This ensures that the expression remains defined.
Cancel Common Factors: Once you have factored the numerator and denominator, cancel out any common factors. Remember that you can only cancel factors, not terms.
Double-Check Your Work: After simplifying, double-check your work to ensure that you have factored correctly and canceled all common factors.
Practice Regularly: The more you practice simplifying rational expressions, the more comfortable and confident you will become with the process.

Real-World Applications

Simplifying rational expressions is not just an abstract mathematical exercise; it has practical applications in various fields. For example, in physics, rational expressions can be used to describe the relationship between variables in equations of motion. In engineering, they can appear in circuit analysis and control systems. In economics, they can be used to model supply and demand curves.

By mastering the skill of simplifying rational expressions, you are not only preparing for math exams but also equipping yourself with a valuable tool for solving real-world problems in science, technology, engineering, and mathematics (STEM) fields.

Conclusion

In this article, we tackled the question of simplifying the rational expression $\frac{x^2-9}{(x-1)(x-3)}$ when $x \neq 1$ or 3. We learned that the key to simplifying rational expressions is to factor the numerator and denominator completely, identify and cancel common factors, and account for any restrictions on the variable. By following these steps, we correctly simplified the expression to $\frac{x + 3}{x - 1}$ and identified option A as the correct answer.

Understanding the process of simplifying rational expressions is crucial for success in algebra and beyond. It requires a solid foundation in factoring techniques and a careful attention to detail. By mastering this skill, you will be well-equipped to handle more complex mathematical problems and real-world applications.

Remember to practice regularly and apply the tips we discussed in this article. With consistent effort, you will become proficient in simplifying rational expressions and confident in your mathematical abilities. Happy simplifying!