Subtract And Simplify Rational Expressions A Step-by-Step Guide

In mathematics, especially when dealing with algebra, simplifying expressions is a fundamental skill. This guide focuses on simplifying rational expressions involving subtraction. Rational expressions are fractions where the numerator and denominator are polynomials. Subtracting these expressions requires a solid understanding of algebraic principles and techniques. This comprehensive guide will walk you through the process, providing clear steps and explanations to help you master this essential skill. Our initial problem involves subtracting two rational expressions with a common denominator, a scenario that simplifies the process significantly. However, we'll also delve into cases where denominators differ, requiring an additional step of finding a common denominator before subtraction can occur. The ability to simplify rational expressions is not just an abstract mathematical skill; it's crucial in various fields, including calculus, engineering, and physics, where complex equations often need to be simplified for further analysis or computation. This guide aims to equip you with the knowledge and practice needed to confidently tackle such problems. By breaking down each step and providing examples, we'll ensure that you not only understand the mechanics of subtraction but also the underlying principles that make it work. So, whether you're a student grappling with algebra or someone looking to refresh your math skills, this guide offers a structured and accessible approach to mastering the subtraction of rational expressions.

Problem Statement

Let's start with the problem at hand. We are asked to subtract two rational expressions: $\frac{7 x^2+6 x}{x-3}-\frac{25 x+6}{x-3}$. Our goal is to simplify the result as much as possible. This involves combining the expressions into a single fraction and then looking for opportunities to factor and cancel out common terms. This process of simplification is not just about getting to the right answer; it's about understanding the structure of the expression and reducing it to its most basic form. Simplified expressions are easier to work with in subsequent calculations and provide a clearer picture of the relationship between variables. In this particular problem, we're fortunate to have a common denominator, which simplifies the initial steps. However, the real challenge lies in the algebraic manipulation that follows – combining like terms, factoring the resulting polynomial, and identifying any common factors between the numerator and denominator that can be canceled. The simplification process is a critical aspect of algebra, allowing us to transform complex expressions into more manageable forms. It's a skill that's applied across various mathematical disciplines and in practical applications where mathematical models need to be simplified for analysis or computation. As we proceed, we'll highlight the key techniques and strategies for effective simplification, ensuring that you develop a robust understanding of the process.

Step-by-Step Solution

1. Combine the Numerators

Since the denominators are the same $(x-3)$, we can directly subtract the numerators:

$\frac{7 x^2+6 x}{x-3}-\frac{25 x+6}{x-3} = \frac{(7 x^2+6 x) - (25 x+6)}{x-3}$.

This step leverages the fundamental principle of fraction subtraction, where fractions with a common denominator can be combined by performing the subtraction operation on their numerators while keeping the denominator constant. This is a crucial step because it consolidates the two separate rational expressions into a single fraction, making it easier to identify potential simplifications. The parentheses in the expression are important because they remind us to distribute the negative sign correctly across all terms in the second numerator. Neglecting this step can lead to errors in the subsequent simplification. Combining the numerators is often the first step in simplifying rational expressions, but it's important to remember that this only applies when the denominators are the same. If the denominators are different, we must first find a common denominator before we can combine the fractions. This initial step sets the stage for the rest of the solution, and performing it accurately is essential for arriving at the correct answer. In the next steps, we'll focus on simplifying the numerator further by combining like terms and looking for opportunities to factor the resulting polynomial.

2. Simplify the Numerator

Now, let's simplify the numerator by distributing the negative sign and combining like terms:

$\frac{7 x^2+6 x - 25 x - 6}{x-3} = \frac{7 x^2 - 19 x - 6}{x-3}$.

This step is crucial in reducing the complexity of the rational expression. The process involves first distributing the negative sign to each term inside the parentheses, effectively changing the sign of each term in the second numerator. This is a common point of error for students, so it's important to pay close attention to this detail. Once the negative sign is properly distributed, we can then combine like terms, which in this case are the terms with the same power of $x$. Combining like terms involves adding or subtracting their coefficients while keeping the variable part the same. This step not only simplifies the expression but also makes it easier to identify potential factoring opportunities in the next step. The simplified numerator, $7x^2 - 19x - 6$, is a quadratic expression, which is a common type of polynomial encountered in algebra. Simplifying the numerator is a critical step because it prepares the expression for factoring, which is often the key to simplifying rational expressions. By reducing the expression to its simplest form, we make it easier to identify patterns and relationships that might not be immediately apparent in the original expression.

3. Factor the Numerator

Next, we need to factor the quadratic expression in the numerator. We are looking for two binomials that multiply to give $7 x^2 - 19 x - 6$.

After factoring, we get:

$\frac{(7x + 2)(x - 3)}{x-3}$.

Factoring the numerator is a crucial step in simplifying rational expressions because it allows us to identify common factors between the numerator and the denominator. The quadratic expression $7x^2 - 19x - 6$ can be factored into two binomials, and this process often requires some trial and error. We look for two numbers that multiply to the product of the leading coefficient (7) and the constant term (-6), which is -42, and add up to the middle coefficient (-19). These numbers are -21 and 2. Once we find these numbers, we can rewrite the middle term as a sum of two terms and then factor by grouping. This technique allows us to systematically break down the quadratic expression into its factors. The factored form of the numerator, $(7x + 2)(x - 3)$, reveals a common factor with the denominator, which is the key to further simplification. Factoring is a fundamental skill in algebra, and mastering it is essential for working with rational expressions and other algebraic problems. In the next step, we'll see how this factorization allows us to simplify the expression further by canceling out common factors.

4. Cancel Common Factors

Now, we can cancel the common factor of $(x-3)$ from the numerator and the denominator:

$\frac{(7x + 2)(x - 3)}{x-3} = 7x + 2$.

This step is the culmination of the simplification process, where we leverage the factored form of the numerator and denominator to identify and cancel out common factors. Canceling common factors is a valid operation because it's equivalent to dividing both the numerator and the denominator by the same quantity, which doesn't change the value of the expression. However, it's important to note that this cancellation is only valid if the common factor is not equal to zero. In this case, we're canceling the factor $(x - 3)$, which means that the original expression is undefined when $x = 3$. This is a critical consideration when working with rational expressions, as we need to be mindful of the values of the variable that would make the denominator zero, as these values are excluded from the domain of the expression. After canceling the common factor, we're left with a simplified expression, $7x + 2$, which is a linear expression. This simplified form is much easier to work with and provides a clearer representation of the relationship between the variables. Canceling common factors is a key step in simplifying rational expressions, and it often leads to a significantly simpler form of the expression.

Final Answer

Therefore, the simplified result of the subtraction is:

$7x + 2$.

This final answer represents the simplified form of the original expression after performing the subtraction and canceling common factors. It's a linear expression, which is much easier to understand and work with than the original rational expression. The process of simplification has transformed a complex fraction into a simple algebraic expression, making it easier to analyze and use in further calculations. This simplification is not just about finding the correct answer; it's about gaining a deeper understanding of the structure of the expression and how it behaves. The simplified form allows us to quickly evaluate the expression for different values of $x$ and to identify key features of its graph, such as its slope and y-intercept. The ability to simplify rational expressions is a valuable skill in algebra and beyond, as it allows us to reduce complex problems to simpler, more manageable forms. This final answer, $7x + 2$, represents the most simplified form of the original expression, and it's the desired result of the subtraction and simplification process. Understanding the steps involved in arriving at this answer is crucial for mastering the simplification of rational expressions.

Key Concepts

1. Rational Expressions: These are fractions where the numerator and denominator are polynomials.
2. Common Denominator: To add or subtract fractions, they must have the same denominator.
3. Factoring: This involves breaking down a polynomial into its constituent factors, which is crucial for simplifying rational expressions.
4. Canceling Common Factors: This step simplifies the expression by removing factors that appear in both the numerator and denominator.

These key concepts are fundamental to understanding and working with rational expressions. Rational expressions are a core topic in algebra and are used extensively in calculus and other higher-level mathematics. Understanding that rational expressions are essentially fractions with polynomials in the numerator and denominator is the first step towards mastering them. The concept of a common denominator is crucial for adding and subtracting fractions, whether they are simple numerical fractions or more complex rational expressions. Finding a common denominator allows us to combine the fractions into a single expression, which is a necessary step for simplification. Factoring polynomials is another essential skill, as it allows us to identify common factors between the numerator and denominator of a rational expression. Factoring is a process of breaking down a polynomial into its constituent factors, and there are various techniques for factoring different types of polynomials. Finally, canceling common factors is the step that actually simplifies the expression by removing factors that appear in both the numerator and the denominator. This process is valid because it's equivalent to dividing both the numerator and the denominator by the same quantity. Mastering these key concepts is essential for simplifying rational expressions and for success in algebra and beyond.

Additional Tips

• Always look for opportunities to factor the numerator and denominator.
• Be careful when distributing negative signs.
• Remember to state any restrictions on the variable (values that make the denominator zero).

These additional tips are designed to help you avoid common mistakes and improve your problem-solving skills when working with rational expressions. Factoring is often the key to simplifying rational expressions, so always make it a priority to look for opportunities to factor both the numerator and the denominator. This may involve using various factoring techniques, such as factoring out a greatest common factor, factoring by grouping, or using special factoring patterns. Paying close attention to the distribution of negative signs is crucial, especially when subtracting rational expressions. A common mistake is to forget to distribute the negative sign to all terms in the numerator being subtracted, which can lead to an incorrect result. Finally, it's important to remember to state any restrictions on the variable. Rational expressions are undefined when the denominator is equal to zero, so we need to identify and exclude these values from the domain of the expression. This is often done by setting the denominator equal to zero and solving for the variable. By following these tips, you can improve your accuracy and efficiency when simplifying rational expressions and avoid common pitfalls. These tips are not just about getting the right answer; they're about developing good mathematical habits and a deeper understanding of the concepts involved.

By following this guide, you should now have a solid understanding of how to subtract and simplify rational expressions. Remember to practice regularly to reinforce your skills. With practice and a solid understanding of the underlying principles, you'll be able to confidently tackle any problem involving rational expressions.