Factoring: Find Factors Of 6x²y + 8x² - 30y - 40

Oct 14, 2025 by ADMIN 49 views

Hey guys! Let's dive into a factoring problem today. We're going to figure out which of the given options is a factor of the expression $6x^2y + 8x^2 - 30y - 40$ . Factoring can seem tricky, but with a systematic approach, we can break it down and find the solution. So, grab your thinking caps, and let’s get started!

Understanding Factoring

Before we jump into the problem, let's quickly recap what factoring is all about. Factoring, in simple terms, means breaking down an expression into smaller parts (factors) that, when multiplied together, give you the original expression. Think of it like finding the building blocks of a number or an algebraic expression.

For example, if we have the number 12, we can factor it into $2 imes 2 imes 3$ . Similarly, for algebraic expressions, we look for common terms or patterns that allow us to rewrite the expression as a product of factors. This is super useful for simplifying expressions, solving equations, and understanding the behavior of functions. When you master factoring, you unlock a powerful tool in your algebra arsenal.

Common Factoring Techniques

There are several techniques we can use when factoring, but some of the most common ones include:

Greatest Common Factor (GCF): Look for the largest factor that divides all terms in the expression.
Difference of Squares: Recognize patterns like $a^2 - b^2$ , which factors into $(a + b)(a - b)$ .
Perfect Square Trinomials: Identify patterns like $a^2 + 2ab + b^2$ or $a^2 - 2ab + b^2$ , which factor into $(a + b)^2$ or $(a - b)^2$ , respectively.
Factoring by Grouping: Group terms together and factor out common factors from each group.

In this problem, we'll primarily use the technique of factoring by grouping, which is perfect for expressions with four terms. So, let’s keep these techniques in mind as we tackle our specific problem!

Breaking Down the Expression: $6x^2y + 8x^2 - 30y - 40$

Okay, let's get our hands dirty with the expression: $6x^2y + 8x^2 - 30y - 40$ . The key to factoring this kind of expression is recognizing that we have four terms, which often suggests factoring by grouping. Factoring by grouping is a technique where we pair terms, factor out the greatest common factor (GCF) from each pair, and then see if we can find a common binomial factor.

The first step is to group the terms. A natural way to do this is to pair the first two terms and the last two terms together:

$(6x^2y + 8x^2) + (-30y - 40)$

Now, let's look at each group separately and find the GCF. For the first group, $(6x^2y + 8x^2)$ , the GCF is $2x^2$ . For the second group, $(-30y - 40)$ , the GCF is -10. Factoring these out, we get:

$2x^2(3y + 4) - 10(3y + 4)$

Notice anything cool? Both terms now have a common factor of $(3y + 4)$ . This is exactly what we want! This common binomial factor is our ticket to the next step. By recognizing and extracting the GCF from each group, we've made our expression much simpler and set the stage for the final factorization.

Factoring by Grouping: Step-by-Step

Let’s continue factoring our expression. We've reached a crucial point where we have:

$2x^2(3y + 4) - 10(3y + 4)$

As we spotted earlier, the binomial $(3y + 4)$ is common to both terms. This is our golden ticket! To complete the factoring by grouping, we factor out the common binomial factor $(3y + 4)$ from the entire expression. Think of it like pulling out the same Lego brick from two different structures—we’re essentially doing the same thing here.

When we factor out $(3y + 4)$ , we’re left with the terms that were multiplying it, which are $2x^2$ and $-10$ . So, we write:

$(3y + 4)(2x^2 - 10)$

Awesome! We’ve successfully factored by grouping. But hold on, we’re not quite done yet. Always remember to check if you can factor further. In this case, we can see that the second factor, $(2x^2 - 10)$ , has a common factor of 2. Let's factor that out:

$2(x^2 - 5)$

Now, we substitute this back into our expression:

$(3y + 4) imes 2(x^2 - 5)$

So, the fully factored form of the expression is:

$2(3y + 4)(x^2 - 5)$

By methodically factoring out the common binomial and then any remaining numerical factors, we've arrived at the most simplified factored form of our original expression. This step-by-step approach ensures we don’t miss any opportunities to simplify, making our final answer as clean as possible.

Identifying the Correct Factor

Alright, we've successfully factored our expression $6x^2y + 8x^2 - 30y - 40$ into $2(3y + 4)(x^2 - 5)$ . Now, let's circle back to the original question: Which of the following is a factor of this expression?

We were given the following options:

A. $x - 4$ B. $3y + 4$ C. $x^2 + 4$ D. $3y - 5$

Looking at our factored form, $2(3y + 4)(x^2 - 5)$ , we can clearly see which of these options matches one of our factors. The factor $(3y + 4)$ appears directly in our factored expression. This means that option B, $3y + 4$ , is indeed a factor of the original expression.

Options A, C, and D, on the other hand, do not appear in our factored form. This simple comparison highlights the power of factoring: once you break down an expression into its factors, identifying specific factors becomes straightforward. So, in this case, the correct answer is definitely B. Great job, guys! We nailed it.

Why Other Options are Incorrect

To really nail down our understanding, let's briefly discuss why the other options are not factors of the given expression. This helps solidify our factoring skills and ensures we're not just picking the right answer but also understanding why the wrong answers are incorrect.

We factored the expression $6x^2y + 8x^2 - 30y - 40$ and found its factored form to be $2(3y + 4)(x^2 - 5)$ . Let's go through the options one by one:

A. $x - 4$ : This term does not appear anywhere in our factored expression. There's no way to multiply our factors together to get a term like $(x - 4)$ . So, this is incorrect.

C. $x^2 + 4$ : Again, this term isn't present in our factored form. We have $(x^2 - 5)$ , but not $(x^2 + 4)$ . These are different, and one cannot be derived from the other through simple multiplication within our factors.

D. $3y - 5$ : This is close to one of our factors, $(3y + 4)$ , but the sign is different. A change in sign makes it a completely different factor. We cannot obtain $(3y - 5)$ from our factors, so this is also incorrect.

By understanding why these options don't fit, we reinforce our understanding of factoring and how factors work. It's not just about finding the right answer, but also about recognizing why the other possibilities don't work. This deeper understanding is what will help us tackle more complex problems in the future. So, always take that extra moment to analyze why the incorrect options are indeed incorrect – it’s a fantastic learning tool!

Tips and Tricks for Factoring

Factoring can sometimes feel like a puzzle, but with the right strategies, it becomes much more manageable. Here are some tips and tricks to keep in mind as you tackle factoring problems:

Always look for the Greatest Common Factor (GCF) first: This is your first line of defense. Factoring out the GCF simplifies the expression and makes subsequent steps easier.
Recognize patterns: Keep an eye out for patterns like the difference of squares ( $a^2 - b^2$ ) or perfect square trinomials ( $a^2 itle 2ab + b^2$ or $a^2 - 2ab + b^2$ ). Recognizing these patterns can save you a lot of time.
Factoring by grouping for four-term expressions: If you have four terms, try factoring by grouping. Group the terms in pairs and look for common factors in each pair.
Check your work: After factoring, multiply the factors back together to make sure you get the original expression. This is a great way to catch mistakes.
Practice makes perfect: The more you practice factoring, the better you'll become at recognizing patterns and applying the right techniques. Do lots of problems, and don't get discouraged if you don't get it right away.
Stay organized: Write out your steps clearly and keep your work neat. This makes it easier to spot mistakes and follow your reasoning.
Don't give up: Factoring can be challenging, but it's a crucial skill in algebra. If you get stuck, take a break, review the techniques, and try again. You'll get there!

By keeping these tips in mind, you'll be well-equipped to handle a wide range of factoring problems. Factoring is a fundamental skill, and mastering it will open doors to more advanced topics in mathematics. So, keep practicing and stay persistent!

Conclusion: Mastering Factoring

So, guys, we've successfully navigated through a factoring problem! We started with the expression $6x^2y + 8x^2 - 30y - 40$ and, after applying the technique of factoring by grouping, we determined that $(3y + 4)$ is indeed a factor. We didn't just find the answer; we also discussed why the other options were incorrect, which is super important for truly understanding the concepts.

Factoring is a crucial skill in algebra, and it's something you'll use again and again in more advanced math courses. By breaking down complex expressions into simpler parts, we can solve equations, simplify fractions, and gain a deeper understanding of mathematical relationships. The techniques we’ve covered today, like identifying the GCF, recognizing patterns, and factoring by grouping, are all powerful tools in your mathematical toolkit.

Remember, the key to mastering factoring is practice. The more you work through problems, the more comfortable you'll become with the different techniques and patterns. So, keep practicing, stay curious, and don't be afraid to tackle challenging problems. You've got this! And as always, if you ever get stuck, remember to break the problem down into smaller steps, review the basic principles, and maybe even ask a friend or teacher for help. Happy factoring, and I'll catch you in the next math adventure!