Difference In Rational Expressions: A Step-by-Step Analysis

This article delves into the fascinating world of rational expressions, focusing on a specific problem that highlights the nuances of algebraic manipulation and simplification. We will dissect the given expressions, meticulously examining each step involved in finding a common denominator, combining fractions, and ultimately arriving at the simplified form. This comprehensive analysis aims to provide a clear understanding of the underlying principles and techniques, making it easier to tackle similar problems in the future. This is important for anyone studying algebra, calculus, or related fields. Understanding rational expressions is crucial for success in higher-level mathematics.

Unpacking the Initial Problem

We are presented with the following expression, which forms the foundation of our exploration:

$\frac{x}{x^2-2 x-15}-\frac{4}{x^2+2 x-35}$

This expression involves the subtraction of two rational expressions. To effectively address this, our initial task is to factor the denominators. Factoring the denominators allows us to identify common factors and determine the least common denominator (LCD), which is essential for combining the fractions. Let's break down the factorization process step-by-step.

Factoring the Denominators: A Critical First Step

The first denominator, $x^2 - 2x - 15$, is a quadratic expression. We need to find two numbers that multiply to -15 and add up to -2. These numbers are -5 and 3. Therefore, we can factor the first denominator as follows:

$x^2 - 2x - 15 = (x - 5)(x + 3)$

Similarly, let's factor the second denominator, $x^2 + 2x - 35$. We need two numbers that multiply to -35 and add up to 2. These numbers are 7 and -5. Thus, the second denominator can be factored as:

$x^2 + 2x - 35 = (x + 7)(x - 5)$

Now that we have factored both denominators, our original expression can be rewritten as:

$\frac{x}{(x - 5)(x + 3)} - \frac{4}{(x + 7)(x - 5)}$

This factored form is crucial for the next step: finding the least common denominator.

Identifying the Least Common Denominator (LCD)

The least common denominator (LCD) is the smallest expression that is divisible by both denominators. In our case, the denominators are $(x - 5)(x + 3)$ and $(x + 7)(x - 5)$. To find the LCD, we need to include each unique factor with its highest power. The unique factors are $(x - 5)$, $(x + 3)$, and $(x + 7)$. Therefore, the LCD is:

LCD = $(x - 5)(x + 3)(x + 7)$

With the LCD identified, we can now proceed to rewrite each fraction with this common denominator.

Rewriting Fractions with the LCD

To rewrite the first fraction, $\frac{x}{(x - 5)(x + 3)}$, with the LCD, we need to multiply both the numerator and the denominator by the missing factor, which is $(x + 7)$:

$\frac{x}{(x - 5)(x + 3)} * \frac{(x + 7)}{(x + 7)} = \frac{x(x + 7)}{(x - 5)(x + 3)(x + 7)}$

Similarly, for the second fraction, $\frac{4}{(x + 7)(x - 5)}$, we need to multiply both the numerator and the denominator by the missing factor, which is $(x + 3)$:

$\frac{4}{(x + 7)(x - 5)} * \frac{(x + 3)}{(x + 3)} = \frac{4(x + 3)}{(x - 5)(x + 3)(x + 7)}$

Now our expression looks like this:

$\frac{x(x + 7)}{(x - 5)(x + 3)(x + 7)} - \frac{4(x + 3)}{(x - 5)(x + 3)(x + 7)}$

With both fractions having the same denominator, we can now combine them.

Combining the Fractions

To combine the fractions, we subtract the numerators while keeping the common denominator:

$\frac{x(x + 7) - 4(x + 3)}{(x - 5)(x + 3)(x + 7)}$

Next, we need to simplify the numerator by expanding the products and combining like terms.

Simplifying the Numerator

Let's expand the numerator:

$x(x + 7) - 4(x + 3) = x^2 + 7x - 4x - 12$

Now, combine the like terms:

$x^2 + 7x - 4x - 12 = x^2 + 3x - 12$

So, our expression now becomes:

$\frac{x^2 + 3x - 12}{(x - 5)(x + 3)(x + 7)}$

This simplified form of the numerator is crucial for comparing it with the subsequent expressions provided in the problem.

Analyzing the Subsequent Expressions

The problem presents us with a series of expressions that seem to be intermediate steps in the simplification process. Let's examine each one and see how they relate to our derived expression.

Expression 1: $\frac{x^2+3 x+12}{(x-3)(x-5)(x+7)}$

The first expression given is:

$\frac{x^2+3 x+12}{(x-3)(x-5)(x+7)}$

Comparing this to our simplified expression, $\frac{x^2 + 3x - 12}{(x - 5)(x + 3)(x + 7)}$, we immediately notice a difference in the numerator. The constant term is +12 in this expression, whereas it is -12 in our result. Furthermore, the denominator contains $(x-3)$ instead of $(x+3)$. This suggests a potential error in the initial problem statement or in the steps leading to this expression. It is important to double-check each step to identify any mistakes.

Expression 2: $\frac{x(x+3-12)}{(x+3)(x-5)(x+7)}$

The second expression is:

$\frac{x(x+3-12)}{(x+3)(x-5)(x+7)}$

This expression seems to be an attempt to simplify the numerator, but it is unclear how it was derived. The numerator, $x(x + 3 - 12)$, simplifies to $x(x - 9) = x^2 - 9x$. This is significantly different from our numerator, $x^2 + 3x - 12$, and the numerator in the first expression, $x^2 + 3x + 12$. This discrepancy further highlights the need for careful verification of each step in the simplification process. Mathematical accuracy is paramount in these kinds of problems.

Expression 3: $\frac{x^2+3 x+12}{(x+3)(x-5)(x+7)}$

The third expression is:

$\frac{x^2+3 x+12}{(x+3)(x-5)(x+7)}$

This expression has the same numerator as the first expression, $x^2 + 3x + 12$, but the denominator is now $(x+3)(x-5)(x+7)$, which matches the LCD we derived. However, the numerator still differs from our simplified result. This consistency in the numerator between the first and third expressions suggests a possible consistent error in the original steps, particularly when combining the fractions or simplifying the numerator.

Expression 4: $\frac{x^2+3 x-12}{(x+3)(x-5)(x+7)}$

The fourth expression is:

$\frac{x^2+3 x-12}{(x+3)(x-5)(x+7)}$

This expression matches our simplified result. The numerator, $x^2 + 3x - 12$, and the denominator, $(x+3)(x-5)(x+7)$, are exactly what we obtained after correctly combining the fractions and simplifying. This confirms that our step-by-step approach and calculations were accurate.

Identifying the Key Difference and Potential Errors

The key difference lies in the numerator of the expressions. Our correct simplification yields a numerator of $x^2 + 3x - 12$, while some of the provided expressions have a numerator of $x^2 + 3x + 12$. This difference of 24 in the constant term suggests a potential error in handling the constant terms during the simplification process, specifically when expanding and combining like terms after finding the common denominator. The expression with the numerator $x(x+3-12)$ indicates a possible misunderstanding of how to combine terms or a mistake in algebraic manipulation.

Root Cause Analysis of the Discrepancies

To pinpoint the exact error, it would be necessary to retrace the steps from the beginning, paying close attention to the distribution of the negative sign and the combination of like terms. A common mistake in such problems is incorrectly distributing the negative sign when subtracting fractions. For example, in our case, the subtraction of $\frac{4(x + 3)}{(x - 5)(x + 3)(x + 7)}$ involves distributing the -4 across both terms inside the parentheses, which is where a sign error could easily occur. Careful attention to detail is crucial to prevent such mistakes.

Conclusion: The Importance of Precision in Algebraic Manipulation

In conclusion, this detailed analysis highlights the importance of precision and careful execution in algebraic manipulation. The original problem presented a subtraction of rational expressions, which required factoring denominators, finding a common denominator, combining fractions, and simplifying the resulting expression. By systematically working through each step, we arrived at the correct simplified form, $\frac{x^2 + 3x - 12}{(x - 5)(x + 3)(x + 7)}$. The discrepancies in the provided expressions underscore the significance of verifying each step and paying close attention to details, especially when dealing with negative signs and combining like terms. This exercise serves as a valuable lesson in the necessity of accuracy and thoroughness in mathematical problem-solving. This skill is essential for success in more advanced mathematical studies.