Factor 8x³y - 8x²y - 30xy: A Step-by-Step Guide

by Rajiv Sharma 48 views

Introduction

In this comprehensive guide, we're going to dive deep into the world of factoring algebraic expressions, specifically focusing on the expression 8x³y - 8x²y - 30xy. Factoring is a fundamental skill in algebra, and mastering it opens doors to solving more complex equations and simplifying mathematical problems. So, whether you're a student grappling with homework or just someone looking to brush up on your algebra, you've come to the right place. We'll break down each step, making it easy to understand and apply. Let's get started and unlock the secrets of this expression!

Step 1: Identifying the Greatest Common Factor (GCF)

The very first step in factoring any algebraic expression is to identify the greatest common factor (GCF). What's the GCF, you ask? Well, it's the largest factor that divides evenly into all terms of the expression. Think of it as finding the common ground between the terms. In our case, we have three terms: 8x³y, -8x²y, and -30xy. Let's break down each term and see what they have in common. First, consider the coefficients: 8, -8, and -30. The GCF of these numbers is 2, since 2 is the largest number that divides evenly into all three. Now, let's look at the variables. We have , , and x. The GCF here is x, as it's the highest power of x present in all terms. Lastly, we have y in all terms, so y is also part of the GCF. Combining these, the GCF of the entire expression is 2xy. This GCF is the key to simplifying our expression and making it easier to factor further. By identifying and factoring out the GCF, we're essentially peeling back the layers of the expression to reveal its underlying structure. This step is not only crucial for factoring but also for simplifying expressions in general, making it a valuable skill in algebra and beyond. So, always remember to start by looking for the GCF – it's the cornerstone of successful factoring.

Step 2: Factoring out the GCF

Now that we've identified our greatest common factor (GCF) as 2xy, it's time to put it to work. Factoring out the GCF is like reverse distribution – we're essentially dividing each term in the expression by the GCF and writing it outside the parentheses. Remember, our expression is 8x³y - 8x²y - 30xy. We're going to divide each term by 2xy and see what's left. Let's start with the first term, 8x³y. When we divide 8x³y by 2xy, we get 4x². Think of it as (8/2) * (x³/x) * (y/y) = 4x². Next up is the second term, -8x²y. Dividing -8x²y by 2xy gives us -4x. Again, this is (-8/2) * (x²/x) * (y/y) = -4x. Finally, we have the third term, -30xy. Dividing -30xy by 2xy results in -15. This is because (-30/2) * (x/x) * (y/y) = -15. Now, we rewrite the expression with the GCF factored out: 2xy(4x² - 4x - 15). This is a significant step forward. We've taken a complex expression and simplified it, making it easier to handle. The expression inside the parentheses, 4x² - 4x - 15, is a quadratic trinomial, which we'll tackle in the next step. Factoring out the GCF is a powerful technique that not only simplifies expressions but also sets the stage for further factorization. It's like laying the foundation for a building – without it, the rest of the structure can't stand. So, always remember to look for that GCF and factor it out first. It will make your life much easier!

Step 3: Factoring the Quadratic Trinomial

Okay, guys, now we've reached the heart of the problem: factoring the quadratic trinomial 4x² - 4x - 15. This might seem a bit daunting at first, but don't worry, we'll break it down step by step. A quadratic trinomial is an expression of the form ax² + bx + c, where a, b, and c are constants. In our case, a = 4, b = -4, and c = -15. There are several methods to factor a quadratic trinomial, but one of the most common is the 'ac method', which we'll use here. The first step in the ac method is to multiply a and c. So, 4 * -15 = -60. Now, we need to find two numbers that multiply to -60 and add up to b, which is -4. This might take a little trial and error, but let's think about the factors of 60. We have 1 and 60, 2 and 30, 3 and 20, 4 and 15, 5 and 12, and 6 and 10. After a bit of consideration, we see that 6 and 10 are promising. To get a product of -60 and a sum of -4, we need -10 and +6. So, -10 * 6 = -60 and -10 + 6 = -4. Great! We found our numbers. Now, we rewrite the middle term, -4x, using these two numbers: 4x² + 6x - 10x - 15. Notice that we've simply split the -4x into 6x and -10x. The next step is to factor by grouping. We group the first two terms and the last two terms: (4x² + 6x) + (-10x - 15). Now, we factor out the GCF from each group. From the first group, 4x² + 6x, the GCF is 2x. Factoring this out gives us 2x(2x + 3). From the second group, -10x - 15, the GCF is -5. Factoring this out gives us -5(2x + 3). Now, we have 2x(2x + 3) - 5(2x + 3). Notice that both terms have a common factor of (2x + 3). We factor this out: (2x + 3)(2x - 5). And there you have it! We've successfully factored the quadratic trinomial 4x² - 4x - 15 into (2x + 3)(2x - 5). This process might seem long, but with practice, it becomes second nature. The ac method is a powerful tool for factoring quadratic trinomials, and mastering it will significantly boost your algebra skills.

Step 4: Combining the Factors

Alright, we're in the home stretch now! We've done the heavy lifting by factoring out the GCF and factoring the quadratic trinomial. Now, it's time to put all the pieces together and write out the fully factored expression. Remember, in Step 2, we factored out the GCF, 2xy, from the original expression, 8x³y - 8x²y - 30xy, and we got 2xy(4x² - 4x - 15). Then, in Step 3, we factored the quadratic trinomial 4x² - 4x - 15 into (2x + 3)(2x - 5). Now, we simply combine these factors. We take the GCF, 2xy, and multiply it by the factored quadratic trinomial, (2x + 3)(2x - 5). This gives us the fully factored expression: 2xy(2x + 3)(2x - 5). And that's it! We've successfully factored the original expression, 8x³y - 8x²y - 30xy, into its simplest form. This final step is crucial because it presents the expression in a way that reveals its underlying structure. Factoring allows us to see the building blocks of the expression, which can be incredibly useful in solving equations, simplifying fractions, and understanding mathematical relationships. Combining the factors is like completing a puzzle – you've gathered all the pieces, and now you fit them together to see the whole picture. So, always make sure to combine all the factors you've found to get the complete factored expression. It's the final flourish that demonstrates your mastery of factoring!

Conclusion

So there you have it, guys! We've successfully factored the expression 8x³y - 8x²y - 30xy into 2xy(2x + 3)(2x - 5). We started by identifying the greatest common factor (GCF), then factored it out. Next, we tackled the quadratic trinomial using the ac method. Finally, we combined all the factors to get the fully factored expression. Factoring can seem tricky at first, but with practice and a step-by-step approach, it becomes a powerful tool in your algebraic arsenal. Remember to always look for the GCF first, and then use appropriate methods to factor the remaining expression. Factoring is not just a mathematical exercise; it's a way of understanding the structure of algebraic expressions and simplifying complex problems. By mastering factoring, you'll not only improve your algebra skills but also develop a deeper appreciation for the beauty and elegance of mathematics. So, keep practicing, keep exploring, and keep factoring! You've got this!