Simplifying Rational Expressions And Finding Restrictions On Variables

In mathematics, simplifying rational expressions is a fundamental skill, particularly in algebra and calculus. A rational expression is essentially a fraction where the numerator and denominator are polynomials. Simplifying these expressions involves factoring, canceling common factors, and identifying restrictions on the variable. This process is crucial for solving equations, understanding function behavior, and tackling more advanced mathematical concepts.

The problem at hand requires us to simplify the rational expression ${\frac{x^2-10x-24}{x^2-3x-108}}$ and determine the restrictions on the variable ${x}$. Restrictions are values of ${x}$ that would make the denominator equal to zero, which is undefined in mathematics. Therefore, identifying these restrictions is as important as simplifying the expression itself. Let's delve into a detailed step-by-step solution.

Step 1 Factoring the Numerator and Denominator

The first and most crucial step in simplifying any rational expression is to factor both the numerator and the denominator. Factoring allows us to identify common factors that can be canceled out, thereby simplifying the expression. In our case, we have a quadratic expression in both the numerator and the denominator.

The numerator is ${x^2 - 10x - 24}$. We need to find two numbers that multiply to -24 and add up to -10. These numbers are -12 and 2. Thus, we can factor the numerator as follows:

${x^2 - 10x - 24 = (x - 12)(x + 2)}$

Next, we factor the denominator, which is ${x^2 - 3x - 108}$. We need to find two numbers that multiply to -108 and add up to -3. These numbers are -12 and 9. Therefore, the denominator can be factored as:

${x^2 - 3x - 108 = (x - 12)(x + 9)}$

By factoring both the numerator and the denominator, we have transformed the original rational expression into:

${\frac{(x - 12)(x + 2)}{(x - 12)(x + 9)}}$

Factoring quadratic expressions is a core algebraic skill, and mastering it is essential for simplifying rational expressions efficiently. Understanding the relationship between the coefficients and the factors allows for quick and accurate factorization, which is a significant advantage in solving mathematical problems.

Step 2 Canceling Common Factors

After factoring the numerator and the denominator, the next step is to identify and cancel out any common factors. This process simplifies the expression by reducing it to its lowest terms. In our factored expression:

${\frac{(x - 12)(x + 2)}{(x - 12)(x + 9)}}$

We can see that the factor ${(x - 12)}$ appears in both the numerator and the denominator. This common factor can be canceled out, provided that ${x}$ is not equal to 12, as this would make the factor equal to zero, and division by zero is undefined. Canceling this common factor, we get:

${\frac{(x + 2)}{(x + 9)}}$

This simplified expression is much easier to work with and represents the original expression in its simplest form. However, it's crucial to remember the restrictions imposed during the cancellation process. The original expression and the simplified expression are equivalent for all values of ${x}$ except those that make the denominator of the original expression equal to zero.

Identifying and canceling common factors is a crucial step in simplifying rational expressions. It reduces the complexity of the expression, making it easier to analyze and use in further calculations. However, it is equally important to keep track of any restrictions imposed during this process to ensure the equivalence of the original and simplified expressions.

Step 3 Identifying Restrictions on the Variable

Identifying restrictions on the variable is a critical step in simplifying rational expressions. Restrictions are values of the variable that make the denominator of the original expression equal to zero. Division by zero is undefined in mathematics, so these values must be excluded from the domain of the expression. To find the restrictions, we set the original denominator equal to zero and solve for ${x}$.

The original denominator was ${x^2 - 3x - 108}$, which we factored as ${(x - 12)(x + 9)}$. Setting this equal to zero gives us:

${(x - 12)(x + 9) = 0}$

This equation is satisfied when either ${(x - 12) = 0}$ or ${(x + 9) = 0}$. Solving these equations, we find:

${x = 12 \text{ or } x = -9}$

Therefore, the restrictions on the variable ${x}$ are ${x \neq 12}$ and ${x \neq -9}$. These values must be excluded from the domain of the expression to avoid division by zero.

The restrictions are just as important as the simplified expression itself. They tell us the values for which the expression is not defined. When working with rational expressions, always identify the restrictions to ensure the validity of your solutions and calculations.

Step 4 Presenting the Simplified Expression and Restrictions

After simplifying the rational expression and identifying the restrictions on the variable, the final step is to present the results clearly and concisely. This involves stating the simplified expression along with the restrictions. In our case, we have simplified the expression to:

${\frac{x + 2}{x + 9}}$

And we have identified the restrictions as:

${x \neq 12, x \neq -9}$

Therefore, the complete answer is:

${\frac{x + 2}{x + 9}, x \neq 12, x \neq -9}$

This notation clearly states the simplified form of the rational expression and the values of ${x}$ for which the expression is defined. Presenting the results in this manner ensures clarity and completeness, which are essential in mathematical communication.

Conclusion

In conclusion, simplifying the rational expression ${\frac{x^2-10x-24}{x^2-3x-108}}$ involves several key steps: factoring the numerator and denominator, canceling common factors, and identifying restrictions on the variable. By following these steps, we found that the simplified expression is ${\frac{x + 2}{x + 9}}$, with restrictions ${x \neq 12}$ and ${x \neq -9}$. This process highlights the importance of algebraic manipulation skills, particularly factoring, and the necessity of considering restrictions to ensure mathematical accuracy. Mastering these techniques is crucial for success in algebra and higher-level mathematics. Understanding these concepts thoroughly will allow you to confidently tackle similar problems and excel in your mathematical journey.

Simplifying rational expressions is not just a mechanical process; it's a fundamental skill that underpins many areas of mathematics. The ability to manipulate algebraic expressions, factor polynomials, and identify restrictions is crucial for solving equations, analyzing functions, and tackling more advanced mathematical concepts. By understanding the underlying principles and practicing regularly, you can develop a strong foundation in this area and enhance your overall mathematical proficiency.

Therefore, the correct answer is B. ${\frac{x+2}{x+9}, x \neq-9, x \neq 12}$.

To further clarify the process of simplifying rational expressions and addressing common queries, here are some frequently asked questions:

1. What is a rational expression?

A rational expression is a fraction where the numerator and the denominator are polynomials. For example, ${\frac{x^2 + 3x + 2}{x - 1}}$ is a rational expression. Rational expressions are analogous to rational numbers, which are fractions where the numerator and denominator are integers. Understanding rational expressions is crucial in algebra and calculus, as they appear frequently in various mathematical problems and applications.

2. Why do we need to simplify rational expressions?

Simplifying rational expressions makes them easier to work with. A simplified expression is in its lowest terms, which means there are no common factors in the numerator and denominator. This simplification is essential for performing operations such as addition, subtraction, multiplication, and division of rational expressions. Moreover, simplified expressions are easier to analyze and interpret, making it simpler to solve equations, understand function behavior, and tackle more advanced mathematical problems. For instance, when solving an equation involving rational expressions, simplifying them first can significantly reduce the complexity of the problem and the likelihood of making errors.

3. How do I factor polynomials?

Factoring polynomials is a key step in simplifying rational expressions. There are several techniques for factoring polynomials, including:

• Common Factoring: Look for a common factor in all terms of the polynomial and factor it out. For example, ${2x^2 + 4x = 2x(x + 2)}$.
• Factoring by Grouping: Group terms in pairs and factor out common factors from each pair. For example, ${x^3 + 2x^2 + 3x + 6 = x^2(x + 2) + 3(x + 2) = (x^2 + 3)(x + 2)}$.
• Factoring Quadratic Trinomials: Use the quadratic formula or find two numbers that multiply to the constant term and add up to the coefficient of the linear term. For example, ${x^2 + 5x + 6 = (x + 2)(x + 3)}$.
• Difference of Squares: Recognize patterns like ${a^2 - b^2 = (a - b)(a + b)}$. For example, ${x^2 - 9 = (x - 3)(x + 3)}$.
• Sum and Difference of Cubes: Recognize patterns like ${a^3 + b^3 = (a + b)(a^2 - ab + b^2)}$ and ${a^3 - b^3 = (a - b)(a^2 + ab + b^2)}$. For example, ${x^3 + 8 = (x + 2)(x^2 - 2x + 4)}$.

Mastering these techniques is crucial for efficiently simplifying rational expressions. Practice factoring various types of polynomials to build your skills and confidence.

4. What are restrictions on the variable, and why are they important?

Restrictions on the variable are values that make the denominator of the rational expression equal to zero. Division by zero is undefined in mathematics, so these values must be excluded from the domain of the expression. To find the restrictions, set the denominator equal to zero and solve for the variable. For example, in the expression ${\frac{1}{x - 2}}$, the restriction is ${x \neq 2}$.

Restrictions are important because they ensure the mathematical validity of the expression. The original and simplified expressions are equivalent for all values of the variable except those that are restricted. Failing to identify and state the restrictions can lead to incorrect solutions and misunderstandings of the expression's behavior. Always consider the restrictions when working with rational expressions to ensure accurate and meaningful results.

5. How do I cancel common factors in a rational expression?

To cancel common factors, first factor both the numerator and the denominator. Then, identify any factors that appear in both the numerator and the denominator. These common factors can be canceled out, provided that the variable does not take on values that make these factors equal to zero. For example, in the expression ${\frac{(x + 1)(x - 2)}{(x - 2)(x + 3)}}$, the common factor ${(x - 2)}$ can be canceled out, provided that ${x \neq 2}$. Canceling common factors simplifies the expression, making it easier to work with and analyze.

6. What happens if I forget to state the restrictions?

Forgetting to state the restrictions can lead to mathematical inaccuracies. While the simplified expression is equivalent to the original for most values of the variable, it is not equivalent for the values that are restricted. If you forget to state the restrictions, you might inadvertently include these values in the solution set, leading to incorrect results. Additionally, in more advanced mathematical contexts, such as calculus, failing to consider restrictions can result in misinterpretations of function behavior and incorrect calculations of limits and derivatives. Always remember to identify and state the restrictions when simplifying rational expressions to ensure mathematical accuracy and completeness.

7. Can a rational expression have more than one restriction?

Yes, a rational expression can have multiple restrictions. This typically occurs when the denominator has multiple factors that can equal zero. For example, in the expression ${\frac{1}{(x - 1)(x + 2)}}$, the restrictions are ${x \neq 1}$ and ${x \neq -2}$. Each factor in the denominator contributes a potential restriction, so it's crucial to consider all factors when identifying the restrictions on the variable. Rational expressions with multiple restrictions often arise in more complex algebraic problems and applications, making it essential to understand how to identify and state all restrictions correctly.

8. How do I check if my simplified expression is correct?

To check if your simplified expression is correct, you can substitute a few values for the variable in both the original and simplified expressions. If the expressions yield the same result for these values (excluding the restricted values), then your simplification is likely correct. Additionally, you can multiply the simplified expression by the canceled factors to see if you obtain the original expression. This reverse process can help verify that you have correctly factored and canceled common factors. Always perform these checks to ensure the accuracy of your simplified expressions.

By understanding these FAQs, you can enhance your proficiency in simplifying rational expressions and tackling related mathematical problems with greater confidence and accuracy.