Subtracting Rational Expressions A Step-by-Step Guide

In mathematics, working with rational expressions often involves performing operations such as addition, subtraction, multiplication, and division. This article focuses on the subtraction of two rational expressions. Specifically, we will delve into the process of finding the difference between ${\frac{7x}{3x^2 - 13x + 4}}$ and ${\frac{5x}{5x^2 - 22x + 8}}$. This involves factoring the denominators, finding a common denominator, adjusting the numerators, and simplifying the resulting expression. Mastering these steps is crucial for students and anyone involved in mathematical fields, as it forms a fundamental skill applicable in calculus, algebra, and various engineering disciplines. Rational expressions are pervasive in mathematical models, from describing rates of change to modeling physical systems, making their manipulation a key competence.

The initial step in subtracting rational expressions is to factor the denominators. This allows us to identify common factors and determine the least common denominator (LCD). We start by factoring the first denominator, ${3x^2 - 13x + 4}$. To factor this quadratic expression, we look for two numbers that multiply to ${3 \times 4 = 12}$ and add up to -13. These numbers are -12 and -1. Therefore, we can rewrite the middle term as ${-13x = -12x - x}$.

\begin{align*} 3x^2 - 13x + 4 &= 3x^2 - 12x - x + 4 \ &= 3x(x - 4) - 1(x - 4) \ &= (3x - 1)(x - 4) \end{align*}

Next, we factor the second denominator, ${5x^2 - 22x + 8}$. We need two numbers that multiply to ${5 \times 8 = 40}$ and add up to -22. These numbers are -20 and -2. Rewriting the middle term gives us:

\begin{align*} 5x^2 - 22x + 8 &= 5x^2 - 20x - 2x + 8 \ &= 5x(x - 4) - 2(x - 4) \ &= (5x - 2)(x - 4) \end{align*}

By factoring the denominators, we have transformed the original expressions into ${\frac{7x}{(3x - 1)(x - 4)}}$ and ${\frac{5x}{(5x - 2)(x - 4)}}$. This is a critical step because it allows us to identify common factors, which are essential for finding the least common denominator. The ability to correctly factor quadratic expressions is a fundamental skill in algebra, enabling simplification of more complex rational functions and equations. This skill not only aids in solving algebraic problems but also in understanding the structure and behavior of mathematical expressions, which is crucial in higher mathematics and various scientific applications.

After factoring the denominators, the next crucial step is to determine the least common denominator (LCD). The LCD is the smallest multiple that both denominators can divide into evenly. This is vital for combining rational expressions through addition or subtraction. From the previous step, we have the factored denominators as ${(3x - 1)(x - 4)}$ and ${(5x - 2)(x - 4)}$.

To find the LCD, we identify all unique factors present in the denominators and take the highest power of each factor. In this case, the unique factors are ${(3x - 1)}$, ${(x - 4)}$, and ${(5x - 2)}$. Each factor appears only once in each denominator, so we take each factor as is. Thus, the LCD is the product of these factors:

${ \text{LCD} = (3x - 1)(x - 4)(5x - 2) }$

Understanding and correctly determining the LCD is fundamental because it allows us to rewrite each fraction with a common denominator, making subtraction feasible. Without the LCD, subtracting fractions would be akin to comparing apples and oranges—the denominators must be the same to perform the subtraction. The LCD not only simplifies the arithmetic but also provides a structured way to combine expressions, which is essential in more advanced mathematical manipulations. This step is a cornerstone in rational expression manipulation, bridging the gap between basic algebraic fractions and more complex rational functions encountered in calculus and other higher-level mathematics courses.

Now that we have determined the least common denominator (LCD), which is ${(3x - 1)(x - 4)(5x - 2)}$, we need to rewrite each fraction with this LCD. This involves multiplying both the numerator and the denominator of each fraction by the factors that are missing from its original denominator but are present in the LCD.

For the first fraction, ${\frac{7x}{(3x - 1)(x - 4)}}$, the denominator is missing the factor ${(5x - 2)}$. Therefore, we multiply both the numerator and the denominator by ${(5x - 2)}$:

${ \frac{7x}{(3x - 1)(x - 4)} \times \frac{(5x - 2)}{(5x - 2)} = \frac{7x(5x - 2)}{(3x - 1)(x - 4)(5x - 2)} }$

For the second fraction, ${\frac{5x}{(5x - 2)(x - 4)}}$, the denominator is missing the factor ${(3x - 1)}$. We multiply both the numerator and the denominator by ${(3x - 1)}$:

${ \frac{5x}{(5x - 2)(x - 4)} \times \frac{(3x - 1)}{(3x - 1)} = \frac{5x(3x - 1)}{(5x - 2)(x - 4)(3x - 1)} }$

By rewriting the fractions with the LCD, we have now expressed both fractions in a form where they can be directly subtracted. This step is crucial as it standardizes the denominators, allowing for the numerators to be combined. The ability to manipulate fractions in this manner is a fundamental skill in algebra, serving as a stepping stone to more complex algebraic manipulations and problem-solving scenarios. Understanding this process is not just about changing the appearance of the fractions; it’s about creating equivalent expressions that facilitate mathematical operations, which is a core concept in algebra and calculus.

With both fractions now expressed with the least common denominator (LCD), which is ${(3x - 1)(x - 4)(5x - 2)}$, we can proceed to subtract the fractions. The process involves subtracting the numerators while keeping the common denominator. We have:

${ \frac{7x(5x - 2)}{(3x - 1)(x - 4)(5x - 2)} - \frac{5x(3x - 1)}{(5x - 2)(x - 4)(3x - 1)} }$

Subtracting the numerators, we get:

${ \frac{7x(5x - 2) - 5x(3x - 1)}{(3x - 1)(x - 4)(5x - 2)} }$

Now, we expand the terms in the numerator:

${ \frac{35x^2 - 14x - (15x^2 - 5x)}{(3x - 1)(x - 4)(5x - 2)} }$

Distribute the negative sign and combine like terms:

${ \frac{35x^2 - 14x - 15x^2 + 5x}{(3x - 1)(x - 4)(5x - 2)} = \frac{20x^2 - 9x}{(3x - 1)(x - 4)(5x - 2)} }$

This step is the heart of the problem where the actual subtraction takes place. By correctly expanding and simplifying the numerator, we are essentially combining like terms to arrive at a more manageable expression. The skill of subtracting rational expressions is not just a mechanical process; it demonstrates an understanding of algebraic manipulation and the ability to apply fundamental rules of arithmetic in a more complex setting. This is a vital skill for anyone advancing in mathematics, particularly in fields like calculus, where the simplification of complex expressions is a routine task. Mastery of this step lays a strong foundation for more advanced mathematical problem-solving.

After subtracting the fractions, we obtained the expression:

${ \frac{20x^2 - 9x}{(3x - 1)(x - 4)(5x - 2)} }$

The next step is to simplify this expression if possible. We start by factoring the numerator:

${ 20x^2 - 9x = x(20x - 9) }$

Now, we check if any factors in the numerator can cancel with factors in the denominator. The expression becomes:

${ \frac{x(20x - 9)}{(3x - 1)(x - 4)(5x - 2)} }$

By examining the numerator and the denominator, we can see that there are no common factors that can be canceled. Therefore, the simplified form of the expression is:

${ \frac{x(20x - 9)}{(3x - 1)(x - 4)(5x - 2)} }$

This is the final simplified form of the difference between the two original rational expressions. Simplifying the result is a critical step because it presents the solution in its most concise and understandable form. Factoring and canceling common factors not only simplifies the expression but also reduces the complexity of future calculations involving this expression. The ability to recognize and execute simplification is a cornerstone of mathematical proficiency, and it is particularly crucial in fields where complex equations and expressions are commonplace. This final step ties together all the preceding operations, demonstrating a comprehensive understanding of rational expression manipulation and algebraic simplification.

In conclusion, finding the difference between the rational expressions ${\frac{7x}{3x^2 - 13x + 4}}$ and ${\frac{5x}{5x^2 - 22x + 8}}$ involved several key steps: factoring the denominators, determining the least common denominator, rewriting each fraction with the LCD, subtracting the fractions, and simplifying the result. The final simplified expression is:

${ \frac{x(20x - 9)}{(3x - 1)(x - 4)(5x - 2)} }$

Each of these steps is essential for mastering operations with rational expressions. Factoring allows us to identify common factors and find the LCD, which is crucial for combining fractions. Rewriting fractions with the LCD ensures that we can perform subtraction correctly. Simplifying the result presents the answer in its most concise form, making it easier to work with in future calculations. Understanding these processes is not just about following a set of rules; it’s about developing a deep understanding of algebraic manipulation. This understanding is invaluable in various fields, from engineering to computer science, where rational expressions and algebraic simplification are frequently encountered. The ability to systematically approach and solve such problems is a testament to one's mathematical acumen and problem-solving skills. By mastering these techniques, individuals can confidently tackle more complex mathematical challenges, making it a cornerstone of mathematical education.