Factorizing 10xy - 12 + 15x - 8y A Step-by-Step Guide

Introduction

In this comprehensive guide, we will delve into the process of factorizing the algebraic expression 10xy - 12 + 15x - 8y. Factorization is a fundamental concept in algebra, and mastering it is crucial for simplifying expressions, solving equations, and tackling more advanced mathematical problems. Our focus will be on breaking down the given expression into its constituent factors, revealing the underlying structure, and making it easier to work with. This article aims to provide a step-by-step approach, ensuring a clear understanding of the techniques involved. By the end of this guide, you will be equipped with the knowledge and skills to tackle similar factorization problems with confidence.

Understanding Factorization

Before we dive into the specific problem, let's first clarify what factorization entails. Factorization is the process of expressing a number or an algebraic expression as a product of its factors. In simpler terms, we're looking for components that, when multiplied together, give us the original expression. For instance, consider the number 12. It can be factorized as 2 × 2 × 3, where 2 and 3 are its prime factors. Similarly, in algebra, we aim to break down complex expressions into simpler terms, often by identifying common factors among the terms. This not only simplifies the expression but also helps in solving equations and understanding the relationships between variables. Recognizing common factors and applying the distributive property in reverse are key techniques in factorization. The ability to factorize expressions efficiently is a cornerstone of algebraic manipulation and is essential for success in higher-level mathematics.

Step 1: Rearranging the Terms

The given expression is 10xy - 12 + 15x - 8y. The first step in factorizing this expression is to rearrange the terms to group those with common factors together. This rearrangement doesn't change the value of the expression but makes the common factors more apparent. A strategic rearrangement can significantly simplify the factorization process. In this case, we can group the terms containing 'x' and 'y' together, as they are likely to share common factors. By rearranging, we aim to create pairs of terms that have a clear common factor, which we can then extract. This is a crucial step as it sets the foundation for the subsequent steps in the factorization process. The goal is to create a structure that allows us to apply the distributive property in reverse effectively. Thus, we rewrite the expression as 10xy - 8y + 15x - 12. This rearrangement allows us to identify potential common factors within the first two terms and the last two terms, paving the way for the next step in our factorization journey.

Step 2: Identifying Common Factors

After rearranging the terms as 10xy - 8y + 15x - 12, our next crucial step is to identify the common factors within the grouped terms. This involves carefully examining each pair of terms and determining the factors they share. In the first pair, 10xy and -8y, we can see that both terms have 'y' as a common factor. Additionally, both 10 and 8 are divisible by 2. Therefore, the greatest common factor (GCF) for the first pair is 2y. In the second pair, 15x and -12, the common factor is the numerical value. Both 15 and 12 are divisible by 3, making 3 the GCF for this pair. Identifying these common factors is a critical step because it allows us to extract them and simplify the expression further. This process leverages the distributive property in reverse, which is the core principle behind factorization. By recognizing and extracting these common factors, we are one step closer to expressing the original expression as a product of simpler terms.

Step 3: Factoring by Grouping

Now that we've identified the common factors, we can proceed with factoring by grouping. Taking the expression 10xy - 8y + 15x - 12, we factor out the common factor from the first two terms, which is 2y. This gives us 2y(5x - 4). Next, we factor out the common factor from the last two terms, which is 3. This yields 3(5x - 4). At this stage, our expression looks like 2y(5x - 4) + 3(5x - 4). Notice that we now have a common binomial factor, which is (5x - 4). Factoring by grouping is a powerful technique that allows us to simplify complex expressions by breaking them down into smaller, more manageable parts. The key is to identify the common factors within each group and then extract them, revealing a common binomial factor that can be further factored out. This method is particularly useful when dealing with expressions that do not have a single common factor across all terms but can be grouped in a way that reveals shared factors. The presence of the common binomial factor is a clear indication that we are on the right track in our factorization process.

Step 4: Final Factorization

Having reached the expression 2y(5x - 4) + 3(5x - 4), we can now perform the final factorization step. We observe that the binomial (5x - 4) is common to both terms. This allows us to factor out (5x - 4) from the entire expression. By factoring out the common binomial, we are essentially reversing the distributive property. This process simplifies the expression into a product of two factors, which is the ultimate goal of factorization. When we factor out (5x - 4), we are left with 2y from the first term and 3 from the second term. Combining these, we get the final factored form of the expression. This step demonstrates the elegance and efficiency of factorization, as it transforms a complex expression into a more concise and manageable form. The resulting factored form not only simplifies the expression but also provides valuable insights into its structure and behavior.

Result: The Factored Expression

After completing the factorization process, we arrive at the final factored form of the expression 10xy - 12 + 15x - 8y. By factoring out the common binomial (5x - 4), we obtain the result (5x - 4)(2y + 3). This is the fully factorized form of the given expression. It represents the original expression as a product of two binomial factors, simplifying it significantly. The factored form is not only more concise but also reveals the underlying structure of the expression, making it easier to analyze and work with in various mathematical contexts. This result showcases the power of factorization as a technique for simplifying algebraic expressions and highlighting their inherent components. Understanding how to arrive at this factored form is crucial for solving equations, simplifying rational expressions, and tackling more advanced algebraic problems.

Conclusion

In conclusion, we have successfully factorized the expression 10xy - 12 + 15x - 8y into (5x - 4)(2y + 3). This process involved several key steps: rearranging terms, identifying common factors, factoring by grouping, and finally, extracting the common binomial. Each step is crucial in transforming the original expression into its factored form. Factorization is a fundamental skill in algebra, essential for simplifying expressions, solving equations, and gaining a deeper understanding of mathematical relationships. Mastering these techniques allows for more efficient problem-solving and lays the groundwork for advanced mathematical concepts. The ability to factorize expressions not only simplifies complex calculations but also enhances one's ability to analyze and manipulate mathematical equations. By understanding the underlying principles and practicing these steps, anyone can become proficient in factorization and confidently tackle a wide range of algebraic challenges. This comprehensive guide has provided a clear and step-by-step approach to factorizing the given expression, equipping readers with the knowledge and skills to apply these techniques to similar problems.