Simplifying Algebraic Expressions A Step By Step Guide

In the realm of mathematics, simplification is a fundamental skill, enabling us to express complex expressions in a more manageable and understandable form. This article delves into the simplification of a specific algebraic expression, guiding you through the process step-by-step, and highlighting the underlying principles involved. Our main focus will be on simplifying the expression:

${ \frac{4k+2}{k^2-4} \cdot \frac{k-2}{2k+1} }$

with the ultimate goal of identifying the equivalent simplified form from the given options. Algebraic simplification involves a series of techniques, including factoring, canceling common factors, and applying algebraic identities. Mastering these techniques is crucial for success in algebra and beyond. We will dissect the given expression, apply these techniques meticulously, and arrive at the correct simplified form. Let's embark on this mathematical journey, unraveling the intricacies of algebraic simplification.

Dissecting the Expression: A Step-by-Step Approach

To effectively simplify the given algebraic expression, we need to break it down into smaller, more manageable parts. The expression is a product of two fractions:

${ \frac{4k+2}{k^2-4} \cdot \frac{k-2}{2k+1} }$

Our initial strategy involves factoring the numerators and denominators of both fractions. Factoring allows us to identify common factors that can be canceled out, leading to a simplified expression. The first fraction's numerator, 4k + 2, can be factored by taking out the greatest common factor (GCF), which is 2:

${ 4k + 2 = 2(2k + 1) }$

Moving on to the first fraction's denominator, k^2 - 4, we recognize this as a difference of squares. The difference of squares identity states that a^2 - b^2 = (a + b)(a - b). Applying this identity, we factor k^2 - 4 as:

${ k^2 - 4 = (k + 2)(k - 2) }$

Now, let's focus on the second fraction. The numerator, k - 2, is already in its simplest form and cannot be factored further. The denominator, 2k + 1, is also in its simplest form and cannot be factored further. With the factoring complete, our expression now looks like this:

${ \frac{2(2k+1)}{(k+2)(k-2)} \cdot \frac{k-2}{2k+1} }$

The next crucial step is to identify and cancel out any common factors present in both the numerators and denominators. This process is the heart of simplification, as it eliminates redundant terms and leads us closer to the final answer. By systematically factoring each part of the expression, we've set the stage for a clear and efficient simplification process.

Canceling Common Factors: The Art of Simplification

With the expression factored, the next step in our algebraic simplification journey is to cancel out common factors. This process is akin to weeding out unnecessary elements, leaving behind the core essence of the expression. Our factored expression is:

${ \frac{2(2k+1)}{(k+2)(k-2)} \cdot \frac{k-2}{2k+1} }$

We can observe that the term (2k + 1) appears in both the numerator and the denominator. This common factor can be canceled out, effectively dividing both the numerator and denominator by (2k + 1). Similarly, the term (k - 2) also appears in both the numerator and the denominator. This common factor can also be canceled out. After canceling these common factors, the expression simplifies to:

${ \frac{2}{(k+2)} \cdot \frac{1}{1} }$

which further simplifies to:

${ \frac{2}{k+2} }$

The act of canceling common factors is not merely a mechanical process; it's a testament to the fundamental principles of fraction manipulation. It underscores the idea that dividing both the numerator and denominator by the same non-zero quantity does not alter the value of the fraction. By meticulously identifying and canceling these factors, we've stripped away the superficial complexity of the expression, revealing its underlying simplicity. This step is a crucial bridge between the initial complex form and the final simplified form.

Identifying the Simplified Form: Matching the Options

After the cancellation of common factors, we've arrived at the simplified form of our algebraic expression:

${ \frac{2}{k+2} }$

Now, the final step is to compare this simplified form with the given options and identify the correct match. The options presented are:

• A. ${\frac{4}{2k+1}}$
• B. ${\frac{2}{k-2}}$
• C. ${\frac{2}{2k+1}}$
• D. ${\frac{2}{k+2}}$

By direct comparison, it's evident that our simplified form, ${\frac{2}{k+2}}$ , matches option D. This confirms that option D is the correct simplified form of the original expression. The process of identifying the correct option is not merely a matter of visual matching; it's a validation of the entire simplification process. It confirms that each step, from factoring to canceling, was executed correctly, leading us to the accurate final answer. This step provides a sense of closure and reinforces the understanding of the concepts involved. It highlights the importance of meticulousness and accuracy in mathematical simplification, where a single error can lead to an incorrect result. By successfully matching our simplified form with the correct option, we've demonstrated a comprehensive understanding of algebraic simplification techniques.

Conclusion: The Power of Simplification

In this mathematical exploration, we've successfully navigated the process of simplifying a complex algebraic expression. Starting with the expression:

${ \frac{4k+2}{k^2-4} \cdot \frac{k-2}{2k+1} }$

we meticulously applied the principles of factoring, cancellation of common factors, and algebraic manipulation. We began by dissecting the expression, factoring the numerator and denominator of each fraction. This involved recognizing the difference of squares pattern and applying the appropriate identity. Next, we identified and canceled out common factors, a crucial step in simplifying fractions. This process reduced the complexity of the expression, revealing its underlying structure. Finally, we compared our simplified form with the given options and correctly identified option D, ${\frac{2}{k+2}}$ , as the equivalent simplified expression.

This exercise underscores the power and importance of simplification in mathematics. Simplification not only makes expressions easier to work with but also provides deeper insights into their underlying relationships. It's a skill that transcends specific mathematical topics and is applicable across various fields of study and real-world applications. The ability to simplify complex expressions is a testament to one's understanding of mathematical principles and their ability to apply them effectively. It's a skill that empowers individuals to tackle challenging problems and arrive at elegant solutions. As we conclude this journey of algebraic simplification, let's carry forward the lessons learned and continue to explore the beauty and power of mathematics.