Factor -12 + 15x: Step-by-Step Guide

by Viktoria Ivanova 37 views

Hey guys! Today, we're diving into a crucial concept in algebra: factoring out the common factor. This skill is fundamental for simplifying expressions and solving equations, so let's break it down step-by-step. We'll specifically tackle the expression -12 + 15x, guiding you through the process to choose the correct factored form. Factoring is like reverse distribution, so understanding it thoroughly will give you an edge in tackling more complex algebraic problems. Remember, practice makes perfect, so let's jump right in and master this essential technique!

Understanding the Basics of Factoring

Before we dive into our specific problem, let's make sure we're all on the same page about what factoring actually means. Think of factoring as the opposite of expanding. When we expand, we multiply a term (or terms) across an expression inside parentheses. For example, 3(x + 2) expands to 3x + 6. Factoring, on the other hand, is about finding the common elements within an expression and pulling them out to rewrite it as a product. This simplifies the expression and can make it easier to work with in various mathematical contexts, such as solving equations or simplifying fractions. The key to factoring lies in identifying the greatest common factor (GCF) of the terms involved. This GCF is the largest number (and/or variable) that divides evenly into all terms in the expression. Once you've found the GCF, you can factor it out, leaving you with a simplified expression within the parentheses. Factoring is not just a mathematical trick; it's a powerful tool that helps us understand the structure of expressions and solve problems more efficiently. Mastering this technique opens doors to more advanced algebraic concepts, making it an indispensable skill for any math student.

Identifying the Greatest Common Factor (GCF)

The cornerstone of factoring is finding the Greatest Common Factor, or GCF. So, how do we spot this GCF? First, let’s break down the numbers. Look at the coefficients (the numbers in front of the variables) and constants in your expression. In our case, we have -12 and 15. What's the largest number that divides evenly into both -12 and 15? That's right, it's 3! Now, let's consider the variables. Do both terms have an 'x'? Nope! -12 doesn't have an 'x', so 'x' can't be part of our GCF. If both terms contained 'x', we’d then look for the highest power of 'x' that divides evenly into both. But for this problem, our GCF is simply 3. Remember, the GCF is like the common ground between the terms, the largest piece they both share. Finding it is the first, crucial step in successfully factoring an expression. Once you've identified the GCF, you're ready to take the next step: factoring it out.

Factoring Out the GCF from -12 + 15x

Alright, now that we've nailed down the concept of GCF, let’s get our hands dirty and factor out the GCF from our expression: -12 + 15x. We already determined that the GCF of -12 and 15 is 3. So, what do we do with it? We divide each term in the expression by the GCF and then write the GCF outside a set of parentheses, with the results of the division inside. Let’s break it down:

  • Divide -12 by 3: -12 / 3 = -4
  • Divide 15x by 3: 15x / 3 = 5x

Now, we write the GCF (which is 3) outside the parentheses and the results inside: 3(-4 + 5x). And that’s it! We've successfully factored out the GCF. Think of it like unwrapping a present; we've taken the expression and revealed its factored form. This process not only simplifies the expression but also gives us a deeper understanding of its structure. Always double-check your work by distributing the GCF back into the parentheses. If you get the original expression, you know you've factored correctly. This step-by-step approach makes factoring less daunting and more manageable.

Verifying the Result

Okay, we've factored out the GCF, but how do we know we did it right? This is where verification comes in. It's like a safety net, ensuring we haven't made any sneaky mistakes along the way. The easiest way to verify our factoring is to distribute the GCF back into the parentheses. Remember the distributive property? It's where we multiply the term outside the parentheses by each term inside. So, let's take our factored expression, 3(-4 + 5x), and distribute the 3:

  • 3 * -4 = -12
  • 3 * 5x = 15x

Now, we combine these results: -12 + 15x. Guess what? That's our original expression! This confirms that our factoring is correct. Verification isn't just a formality; it's a critical step in the problem-solving process. It builds confidence in your answer and prevents errors from creeping into more complex calculations. Always make it a habit to verify your factored expressions. It's a small step that can make a big difference in your mathematical journey.

Analyzing the Answer Choices

Now that we've factored the expression and verified our result, let's take a look at the answer choices provided. This is a crucial step, especially in multiple-choice scenarios. We have the following options:

A) 3(-4 + 5x) B) 3x(-4 + 5x) C) x(-12 + 15x) D) x(-8 + 5x^2)

Remember, our factored expression is 3(-4 + 5x). Let's compare this to the answer choices. Option A, 3(-4 + 5x), is a perfect match! This is our correct answer. What about the other options? Option B, 3x(-4 + 5x), incorrectly includes 'x' as part of the GCF. While 3 is the correct numerical GCF, 'x' is not a factor of both terms in the original expression. Option C, x(-12 + 15x), also incorrectly factors out 'x', and it doesn't factor out the numerical GCF of 3. Option D, x(-8 + 5x^2), is way off; it doesn't even come close to the original expression when expanded. Analyzing the answer choices is a great way to solidify your understanding and ensure you've selected the most accurate response. It also helps you identify common mistakes, like incorrectly including variables in the GCF.

Common Factoring Mistakes to Avoid

Factoring can be tricky, and even the best of us make mistakes sometimes. But knowing the common pitfalls can help you steer clear of them. One of the biggest mistakes is failing to identify the greatest common factor. For example, you might factor out 2 from -12 + 15x, getting 2(-6 + 7.5x), but this isn't fully factored because 3 is the GCF. Another common error is incorrectly including a variable in the GCF when it's not present in all terms. We saw this in one of the incorrect answer choices where 'x' was factored out even though -12 doesn't have an 'x'. Sign errors are also frequent culprits. Remember to pay close attention to the signs when dividing each term by the GCF. A negative divided by a positive is negative, and vice versa. And of course, forgetting to verify your answer by distributing can leave errors undetected. Verification is your safety net! By being aware of these common mistakes, you can actively work to avoid them and boost your factoring accuracy. Factoring is a foundational skill, and mastering it requires practice and attention to detail.

Conclusion: Mastering Factoring for Algebraic Success

Alright guys, we've reached the end of our journey into factoring out the common factor from -12 + 15x. We've covered the basics of factoring, learned how to identify the GCF, walked through the steps of factoring it out, verified our result, analyzed answer choices, and even discussed common mistakes to avoid. Whew! That's a lot! But the key takeaway here is that factoring, while it might seem daunting at first, is a skill that can be mastered with practice and a solid understanding of the underlying concepts. Remember, factoring is like reverse distribution, and the GCF is your best friend. It's the key to unlocking the factored form of an expression. By mastering factoring, you're not just learning a mathematical trick; you're building a foundation for more advanced algebraic concepts. So, keep practicing, keep verifying, and keep exploring the wonderful world of algebra! You've got this!

Final Answer: A) 3(-4 + 5x)