Factoring A-B When A=125 And B=27p^12

Hey everyone! Today, we're diving into a fun algebra problem that involves factoring the difference of two monomials. We're given that A = 125 and B = 27p¹², and our mission, should we choose to accept it, is to find the factored form of A - B. Sounds like a blast, right? Let's break it down step-by-step and make sure we understand every nook and cranny of this problem.

Understanding the Problem: A Deep Dive

Before we jump into the solution, let's really get our heads around what we're dealing with. We have two monomials: A and B. A monomial, for those who might need a quick refresher, is simply an algebraic expression consisting of one term. This term can be a number, a variable, or the product of numbers and variables. In our case, A is a constant monomial (125), and B is a monomial involving a variable (p). The key operation we're focusing on is subtraction, specifically finding A - B, which means we're looking at the difference of these two monomials. Factoring, in general, is the process of breaking down an expression into its multiplicative components. Think of it like reverse multiplication. Instead of multiplying things together to get a result, we're starting with the result and figuring out what was multiplied to get it. When we talk about the "factored form" of an expression, we're essentially asking for the expression to be written as a product of simpler expressions (its factors). These simpler expressions, when multiplied together, will give us the original expression. For example, the factored form of x² - 4 is (x + 2)(x - 2). Now, let's bring this back to our specific problem. We want to factor A - B, which is 125 - 27p¹². This looks like it might fit a special factoring pattern, and that's what we'll explore next. Understanding the problem is crucial, guys. It's like having a map before embarking on a journey. If we don't know where we're going, we'll just wander aimlessly. So, take your time to dissect the problem, identify the key components, and understand the goal. In this case, we've established that we're dealing with monomials, a difference operation, and the concept of factoring. With this foundation, we're ready to tackle the solution head-on!

Identifying the Factoring Pattern: Difference of Cubes

Now, let's put our detective hats on and figure out which factoring pattern applies here. The expression 125 - 27p¹² has a specific form that should ring a bell: it looks like a difference of cubes. Remember the difference of cubes pattern? It goes like this: a³ - b³ = (a - b)(a² + ab + b²). This pattern is a powerful tool in our factoring arsenal, and recognizing it is half the battle. So, why do we think this pattern fits our problem? Well, 125 is a perfect cube (5³), and 27p¹² is also a perfect cube! Notice that 27 is 3³, and p¹² can be written as (p⁴)³. This means we can rewrite B as (3p⁴)³. Aha! The pieces are falling into place. We can now see that our expression 125 - 27p¹² can be rewritten as 5³ - (3p⁴)³. This perfectly matches the a³ - b³ form, where a = 5 and b = 3p⁴. Recognizing the pattern is like finding the right key to unlock a door. It allows us to apply a specific formula or technique to simplify the expression. In this case, identifying the difference of cubes pattern gives us a clear roadmap for factoring. Once we've correctly identified the pattern, the rest is just plugging in the values and simplifying. It's like following a recipe once you know all the ingredients and the steps. The difference of cubes pattern isn't the only factoring pattern out there, of course. There's also the sum of cubes, the difference of squares, and various other techniques. The key is to become familiar with these patterns and learn to recognize them in different contexts. Practice makes perfect, guys! The more problems you solve, the quicker you'll be able to spot these patterns and apply the appropriate factoring methods. So, keep those pencils moving and those brains churning!

Applying the Difference of Cubes Formula

Alright, we've identified the pattern – time to put it to work! We know that a = 5 and b = 3p⁴, and our formula is a³ - b³ = (a - b)(a² + ab + b²). Let's substitute these values into the formula. First, we have (a - b), which becomes (5 - 3p⁴). Next, we have (a² + ab + b²). Let's break this down piece by piece: * a² = 5² = 25 * ab = 5 * (3p⁴) = 15p⁴ * b² = (3p⁴)² = 9p⁸ So, (a² + ab + b²) becomes (25 + 15p⁴ + 9p⁸). Now, let's put it all together. Our factored form is (5 - 3p⁴)(25 + 15p⁴ + 9p⁸). And there you have it! We've successfully factored the expression using the difference of cubes formula. The application of the formula is like the execution phase of a plan. We've identified the pattern, gathered our ingredients (the values of a and b), and now we're following the recipe (the formula) to get the final result. Each step in the substitution process is crucial. Make sure you're substituting the correct values and that you're performing the operations in the correct order. A small mistake in the substitution can lead to a completely different answer. Double-checking your work is always a good idea, especially in math! Once you've substituted the values, the next step is simplification. This might involve squaring terms, multiplying expressions, or combining like terms. The goal is to get the expression into its simplest possible form. In our case, we had to square 5, multiply 5 by 3p⁴, and square 3p⁴. These are all basic algebraic operations, but it's important to be careful and methodical. So, guys, remember the key steps in applying a factoring formula: identify the pattern, substitute the values, simplify the expression, and double-check your work. With practice, you'll become a factoring pro in no time!

Checking the Answer: A Crucial Step

We've got our factored form: (5 - 3p⁴)(25 + 15p⁴ + 9p⁸). But before we do a victory dance, let's make absolutely sure we're right. The best way to check our answer is to multiply the factors back together and see if we get our original expression, 125 - 27p¹². This process is like verifying your solution in a puzzle. You've put the pieces together, but you need to make sure they fit perfectly to form the complete picture. Let's multiply (5 - 3p⁴) by (25 + 15p⁴ + 9p⁸). We'll use the distributive property, which means we'll multiply each term in the first parenthesis by each term in the second parenthesis. Here we go: * 5 * 25 = 125 * 5 * 15p⁴ = 75p⁴ * 5 * 9p⁸ = 45p⁸ * -3p⁴ * 25 = -75p⁴ * -3p⁴ * 15p⁴ = -45p⁸ * -3p⁴ * 9p⁸ = -27p¹² Now, let's add all these terms together: 125 + 75p⁴ + 45p⁸ - 75p⁴ - 45p⁸ - 27p¹² Notice anything cool? The 75p⁴ and -75p⁴ cancel each other out, and so do the 45p⁸ and -45p⁸. We're left with 125 - 27p¹². Woo-hoo! That's exactly what we started with. This confirms that our factored form is correct. Checking your answer is a critical step in any math problem, guys. It's like having a safety net. It allows you to catch any mistakes you might have made along the way and correct them before submitting your final answer. It also builds confidence in your solution. When you know you've checked your answer and it's correct, you can be sure that you're on the right track. The multiplication process can sometimes be tedious, especially when dealing with multiple terms. But it's worth the effort. Take your time, be careful with your signs, and double-check each step. A small error in the multiplication can throw off your entire verification process. So, always make checking your answer a habit. It's a sign of a smart and responsible problem-solver!

The Final Answer: A Victory Lap

Drumroll, please! After our careful journey through identifying the pattern, applying the formula, and meticulously checking our answer, we've arrived at the final destination. The factored form of 125 - 27p¹² is indeed (5 - 3p⁴)(25 + 15p⁴ + 9p⁸). Give yourselves a pat on the back, guys! You've tackled a factoring problem like pros. We started with a seemingly complex expression and, by breaking it down into smaller steps, we were able to find its factored form. This is the beauty of algebra! It's like having a set of tools that allow you to solve intricate problems by applying systematic methods. The key takeaways from this problem are: * Recognizing factoring patterns is crucial. The difference of cubes pattern was our key to success in this problem. * Applying the formula correctly is essential. Make sure you substitute the values carefully and perform the operations in the correct order. * Checking your answer is a must. It's the best way to ensure that your solution is correct and that you haven't made any mistakes along the way. Factoring might seem challenging at first, but with practice and a solid understanding of the patterns, you'll become more and more confident. Remember, each problem you solve is a step forward in your mathematical journey. So, keep practicing, keep learning, and keep exploring the amazing world of algebra! And that’s it for today’s adventure in factoring. I hope this breakdown has been helpful and has shed some light on how to approach these types of problems. Remember, the most important thing is to understand the underlying concepts and to practice regularly. Happy factoring, everyone!

So the correct answer is A. (5 - 3p⁴)(25 + 15p⁴ + 9p⁸)