Rewriting Rational Expressions With Common Denominator (x-2)(x-5)(x-6)

In mathematics, especially in algebra, rewriting rational expressions with a common denominator is a fundamental skill. This process is essential for performing operations such as addition and subtraction of fractions with different denominators. This article will delve into the process of rewriting the given rational expressions, $\frac{7}{x^2-7x+10}$ and $\frac{4x}{x^2-8x+12}$, with the common denominator $(x-2)(x-5)(x-6)$. This involves factoring the original denominators, identifying the least common multiple (LCM), and adjusting the numerators accordingly. Understanding these steps will not only help in solving this particular problem but also in tackling a wide range of algebraic manipulations.

When dealing with rational expressions, the ability to manipulate them into equivalent forms is crucial. This skill is not just limited to academic exercises; it is practically applied in various fields such as engineering, physics, and computer science. For instance, in electrical engineering, simplifying complex circuit equations often involves manipulating rational functions. Similarly, in computer graphics, transformations and projections can be expressed using rational expressions. Therefore, mastering the technique of finding common denominators is an investment in a versatile mathematical tool.

Furthermore, this article will emphasize the importance of step-by-step problem-solving. Each step in the process, from factoring the quadratic expressions to multiplying the numerators by the appropriate factors, will be explained in detail. This methodical approach is beneficial not only for accuracy but also for developing a deeper understanding of the underlying principles. By breaking down the problem into manageable parts, we can avoid common mistakes and build confidence in our algebraic skills. This comprehensive guide aims to equip you with the necessary knowledge and techniques to confidently rewrite rational expressions with common denominators.

The first step in rewriting rational expressions with a common denominator is to factor the existing denominators. Factoring helps us identify the individual factors that make up the denominators, which is crucial for finding the least common multiple (LCM). In our case, the denominators are $x^2 - 7x + 10$ and $x^2 - 8x + 12$. Let's factor each of these quadratic expressions.

To factor the first denominator, $x^2 - 7x + 10$, we look for two numbers that multiply to 10 and add up to -7. These numbers are -2 and -5. Thus, we can factor the quadratic as follows:

$x^2 - 7x + 10 = (x - 2)(x - 5)$

Similarly, for the second denominator, $x^2 - 8x + 12$, we need two numbers that multiply to 12 and add up to -8. These numbers are -2 and -6. Therefore, the factored form is:

$x^2 - 8x + 12 = (x - 2)(x - 6)$

Factoring these denominators is a critical step. By breaking down the quadratics into their linear factors, we can clearly see the components that make up each denominator. This makes it easier to identify the least common multiple, which is the smallest expression that is divisible by both denominators. The ability to quickly and accurately factor quadratic expressions is a valuable skill in algebra, and it forms the foundation for many other algebraic manipulations. Understanding the process of factoring not only helps in finding common denominators but also in simplifying rational expressions, solving quadratic equations, and more. Therefore, mastering this technique is essential for anyone studying algebra or related fields.

After factoring the denominators, the next crucial step is to determine the least common multiple (LCM). The LCM is the smallest expression that is divisible by both denominators. In our case, the factored denominators are $(x - 2)(x - 5)$ and $(x - 2)(x - 6)$. To find the LCM, we identify all unique factors present in the denominators and take the highest power of each factor.

The factors in the first denominator are $(x - 2)$ and $(x - 5)$. The factors in the second denominator are $(x - 2)$ and $(x - 6)$. Comparing these, we see that the unique factors are $(x - 2)$, $(x - 5)$, and $(x - 6)$. Each of these factors appears with a power of 1. Therefore, the LCM is the product of these factors:

LCM = $(x - 2)(x - 5)(x - 6)$

This LCM will serve as the common denominator for our rewritten rational expressions. Identifying the LCM correctly is vital because it ensures that we are using the smallest possible expression as the common denominator. This simplifies the subsequent steps and reduces the chances of making errors. The concept of LCM is not only applicable to algebraic expressions but also to integers, making it a fundamental concept in mathematics. Understanding how to find the LCM is essential for adding, subtracting, and simplifying fractions, whether they are numerical or algebraic. Moreover, the LCM plays a significant role in various mathematical contexts, including number theory and polynomial arithmetic. Therefore, mastering the process of finding the LCM is a crucial skill for anyone working with mathematical expressions and equations.

Now that we have the LCM, $(x - 2)(x - 5)(x - 6)$, we can rewrite the given rational expressions with this common denominator. To do this, we need to multiply the numerator and denominator of each fraction by the factors that are missing from its original denominator compared to the LCM.

For the first rational expression, $\frac{7}{x^2 - 7x + 10}$, we know that the denominator factors into $(x - 2)(x - 5)$. To obtain the LCM as the new denominator, we need to multiply both the numerator and the denominator by the missing factor, which is $(x - 6)$:

$\frac{7}{(x - 2)(x - 5)} \times \frac{(x - 6)}{(x - 6)} = \frac{7(x - 6)}{(x - 2)(x - 5)(x - 6)}$

So, the first rewritten expression is $\frac{7(x - 6)}{(x - 2)(x - 5)(x - 6)}$.

Next, we rewrite the second rational expression, $\frac{4x}{x^2 - 8x + 12}$. The denominator factors into $(x - 2)(x - 6)$. To obtain the LCM, we need to multiply both the numerator and the denominator by the missing factor, which is $(x - 5)$:

$\frac{4x}{(x - 2)(x - 6)} \times \frac{(x - 5)}{(x - 5)} = \frac{4x(x - 5)}{(x - 2)(x - 5)(x - 6)}$

Thus, the second rewritten expression is $\frac{4x(x - 5)}{(x - 2)(x - 5)(x - 6)}$.

Rewriting rational expressions with a common denominator is a critical skill in algebra. It allows us to perform operations such as addition and subtraction on fractions with different denominators. The key to this process is to correctly identify the LCM and then multiply the numerators by the appropriate factors. This ensures that the value of the expression remains unchanged while the denominator is transformed to the desired form. This technique is not only useful in algebraic manipulations but also in more advanced mathematical concepts, such as calculus and differential equations. Therefore, mastering this skill is essential for success in higher-level mathematics.

While the expressions $\frac{7(x - 6)}{(x - 2)(x - 5)(x - 6)}$ and $\frac{4x(x - 5)}{(x - 2)(x - 5)(x - 6)}$ are perfectly valid answers, it is often beneficial to expand the numerators to simplify the expressions further or to facilitate additional operations. Expanding the numerators involves multiplying out the factors in the numerator to obtain a polynomial expression.

For the first expression, $\frac{7(x - 6)}{(x - 2)(x - 5)(x - 6)}$, we expand the numerator:

$7(x - 6) = 7x - 42$

So, the first expression becomes $\frac{7x - 42}{(x - 2)(x - 5)(x - 6)}$.

For the second expression, $\frac{4x(x - 5)}{(x - 2)(x - 5)(x - 6)}$, we expand the numerator:

$4x(x - 5) = 4x^2 - 20x$

Thus, the second expression becomes $\frac{4x^2 - 20x}{(x - 2)(x - 5)(x - 6)}$.

Expanding the numerators can be particularly useful when we need to add or subtract these rational expressions. With the numerators in polynomial form, we can easily combine like terms. Additionally, expanding the numerators can sometimes reveal opportunities for further simplification, such as factoring out common factors or canceling terms. However, it is important to note that expanding the numerators is not always necessary, and in some cases, it may make the expression more complex. The decision to expand the numerators depends on the specific context and the operations that need to be performed. In many cases, the expanded form is preferred for clarity and ease of manipulation, especially when dealing with more complex algebraic expressions.

In summary, we have rewritten the given rational expressions with the common denominator $(x-2)(x-5)(x-6)$. The rewritten expressions are:

$\frac{7(x - 6)}{(x - 2)(x - 5)(x - 6)}$ and $\frac{4x(x - 5)}{(x - 2)(x - 5)(x - 6)}$

Optionally, expanding the numerators, we have:

$\frac{7x - 42}{(x - 2)(x - 5)(x - 6)}$ and $\frac{4x^2 - 20x}{(x - 2)(x - 5)(x - 6)}$

These expressions are equivalent to the original expressions but now share a common denominator, which is essential for performing operations such as addition and subtraction. The process involved factoring the original denominators, finding the least common multiple (LCM), and multiplying the numerators by the appropriate factors to match the LCM. This technique is fundamental in algebra and is used extensively in various mathematical applications.

Understanding how to rewrite rational expressions with a common denominator is a crucial skill for anyone studying algebra or related fields. It not only allows for the simplification and manipulation of algebraic expressions but also provides a foundation for more advanced concepts, such as solving rational equations and inequalities. The ability to confidently perform these operations is a key step towards mastering algebraic problem-solving. Furthermore, this process highlights the importance of careful, step-by-step problem-solving, which is a valuable skill in mathematics and beyond. By breaking down complex problems into smaller, manageable steps, we can increase our accuracy and understanding, leading to greater success in mathematical endeavors.

Rewriting rational expressions with a common denominator is a core algebraic skill with wide-ranging applications. This article has demonstrated the process of rewriting $\frac{7}{x^2-7x+10}$ and $\frac{4x}{x^2-8x+12}$ with the common denominator $(x-2)(x-5)(x-6)$. The key steps include factoring the denominators, identifying the least common multiple (LCM), and adjusting the numerators accordingly. By mastering these steps, you can confidently manipulate rational expressions and perform various algebraic operations.

This skill is not only essential for academic purposes but also has practical applications in various fields, including engineering, physics, and computer science. The ability to simplify and manipulate algebraic expressions is crucial for solving complex problems and modeling real-world phenomena. Furthermore, the systematic approach to problem-solving demonstrated in this article is a valuable skill that can be applied to a wide range of challenges. By breaking down complex problems into manageable steps, we can increase our accuracy, understanding, and overall problem-solving abilities.

In conclusion, the ability to rewrite rational expressions with a common denominator is a cornerstone of algebraic proficiency. It enables us to perform operations on rational expressions, simplify complex equations, and solve a variety of mathematical problems. By understanding and practicing this technique, you will be well-equipped to tackle more advanced algebraic concepts and real-world applications. This article serves as a comprehensive guide to this essential skill, providing a step-by-step approach and highlighting the importance of each step in the process. With practice and dedication, you can master this technique and enhance your overall mathematical abilities.