Factor 42xy - 49x + 30y - 35: A Step-by-Step Guide

Hey guys! Let's dive into the fascinating world of factoring and tackle the expression 42xy - 49x + 30y - 35. Factoring, in simple terms, is like reverse multiplication. We're trying to break down a complex expression into simpler pieces (factors) that, when multiplied together, give us the original expression. This skill is super crucial in algebra and beyond, helping us solve equations, simplify expressions, and understand mathematical relationships better. Think of it as cracking a code – we're unraveling the hidden structure within the expression. So, grab your metaphorical magnifying glasses, and let's get started on this factoring adventure!

Understanding the Expression: 42xy - 49x + 30y - 35

Before we jump into factoring, it's essential to understand what we're dealing with. Our expression, 42xy - 49x + 30y - 35, is a polynomial with four terms. Each term consists of coefficients (the numbers) and variables (like x and y). The key to successful factoring is spotting patterns and relationships between these terms. For instance, do you notice any common factors among the coefficients? Or any shared variables between terms? These are the little clues that will guide us on our factoring journey. Recognizing these patterns is like learning the language of algebra; the more fluent we become, the easier it is to manipulate and simplify expressions. Let's take a closer look at each term individually to see what insights we can glean. By dissecting the expression, we're setting ourselves up for success in the next steps.

Grouping Terms: A Strategic Approach

One of the most effective techniques for factoring expressions like ours is grouping. The idea behind grouping is to pair up terms that share common factors, making the factoring process more manageable. In our case, we can group the first two terms (42xy and -49x) and the last two terms (30y and -35). This strategic grouping is like organizing your tools before a big project – it sets the stage for a smoother and more efficient process. When we group terms, we're essentially creating smaller, more digestible chunks that are easier to factor individually. This approach not only simplifies the factoring process but also helps us identify patterns that might not be obvious at first glance. So, let's see how this grouping strategy plays out in our specific expression.

Extracting Common Factors: Unveiling the Hidden Structure

Now that we've grouped our terms, the next step is to extract the greatest common factor (GCF) from each group. Remember, the GCF is the largest factor that divides into all the terms in a group. For the first group (42xy - 49x), the GCF is 7x. Factoring out 7x gives us 7x(6y - 7). For the second group (30y - 35), the GCF is 5. Factoring out 5 gives us 5(6y - 7). This step is like finding the building blocks of our expression – we're uncovering the common elements that link the terms together. Notice anything interesting? Both groups now share a common factor: (6y - 7). This is a crucial observation that will lead us to the final factored form. By extracting common factors, we're not just simplifying the expression; we're also revealing its underlying structure.

The Factored Form: Putting It All Together

Hey! Remember those common factors we identified? That's the key to unlocking the final factored form. We've got 7x(6y - 7) + 5(6y - 7). Notice that (6y - 7) is a common factor to both terms. We can factor this out, just like we did before. This gives us (6y - 7)(7x + 5). Ta-da! We've successfully factored the expression. It's like completing a puzzle – all the pieces fit together perfectly. This factored form is equivalent to our original expression, but it's written in a way that reveals its underlying structure and relationships. Factoring is not just a mathematical trick; it's a way of gaining deeper insights into the nature of expressions and equations.

Verifying the Factors: Ensuring Accuracy

Before we celebrate our factoring success, it's always a good idea to double-check our work. We can do this by multiplying the factors we found, (6y - 7)(7x + 5), and seeing if we get back our original expression, 42xy - 49x + 30y - 35. When we multiply the factors, we get: (6y * 7x) + (6y * 5) + (-7 * 7x) + (-7 * 5) = 42xy + 30y - 49x - 35. And guess what? That matches our original expression! It's like having a cheat code to ensure we've done everything correctly. This verification step is crucial because it helps us catch any mistakes and build confidence in our factoring skills. So, always remember to verify your factors – it's the mark of a true factoring pro!

Analyzing the Options: Which Statements Are True?

Now that we have the factored form, (6y - 7)(7x + 5), we can easily evaluate the given statements:

A. One of the factors is (6y + 7). B. One of the factors is (7x + 5). C. One of the factors is (6y - 7). D. One of the factors is (7x - 5).

By comparing our factored form with the options, we can see that statements B and C are true. (7x + 5) and (6y - 7) are indeed factors of the expression. It's like having the answers right in front of us! This step highlights the power of factoring – it allows us to break down complex expressions and easily identify their components. So, the correct answers are B and C. We've not only factored the expression but also successfully applied our factoring skills to solve a specific problem.

Key Takeaways: Mastering the Art of Factoring

Alright, guys, we've reached the end of our factoring adventure! Let's recap the key takeaways:

• Factoring is like reverse multiplication – breaking down expressions into simpler factors.
• Grouping terms strategically can simplify the factoring process.
• Extracting the greatest common factor (GCF) is crucial for revealing the underlying structure.
• Always verify your factors by multiplying them back together.

Factoring might seem daunting at first, but with practice and the right techniques, it becomes a powerful tool in your mathematical arsenal. Remember, it's not just about getting the right answer; it's about understanding the process and developing your problem-solving skills. So, keep practicing, keep exploring, and keep factoring!

Practice Problems: Sharpening Your Factoring Skills

To solidify your understanding, here are a few practice problems:

1. Factor: 15ab + 20a - 18b - 24
2. Factor: 35pq - 28p + 45q - 36
3. Factor: 12rs - 18r - 10s + 15

Try applying the techniques we discussed – grouping, extracting GCFs, and verifying your factors. Remember, practice makes perfect! The more you factor, the more comfortable and confident you'll become. So, go ahead, challenge yourself, and unlock the power of factoring!

Conclusion: The Power of Factoring

Factoring, my friends, is more than just a mathematical skill; it's a way of thinking. It teaches us to look for patterns, break down complex problems into smaller parts, and appreciate the interconnectedness of mathematical concepts. By mastering factoring, you're not just learning algebra; you're developing critical thinking skills that will serve you well in all areas of life. So, embrace the challenge, enjoy the journey, and keep exploring the fascinating world of mathematics. You've got this! Remember, every factored expression is a testament to your problem-solving prowess. Keep up the great work, and happy factoring!