Simplify (a + B)³ − (a − B)³ − 5a²b: Step-by-Step Solution
Hey guys! Ever stumbled upon an algebraic expression that looks like it belongs in a math Olympics competition? Well, today, we're going to tackle one of those beasts and break it down step by step. We're diving into the simplification of the expression (a + b)³ − (a − b)³ − 5a²b. Sounds intimidating? Don't worry, we'll make it as clear as a sunny day. Let's get started!
Understanding the Challenge
So, our mission, should we choose to accept it (and we do!), is to simplify the expression: (a + b)³ − (a − b)³ − 5a²b. This looks like a complex mix of binomial expansions and subtraction, but trust me, with the right approach, it's totally manageable. We'll need to remember our binomial expansion formulas and be careful with our signs. The goal is to reduce this expression to its simplest form, which will match one of the given alternatives. Before we jump into the solution, let's take a moment to understand why simplifying expressions like this is super important. In mathematics and many fields that use it (like physics, engineering, and computer science), simplified expressions are easier to work with. They help us solve equations, understand relationships between variables, and even make complex calculations more efficient. Think of it like this: a simplified expression is like a well-organized toolbox where you can quickly find the right tool, whereas a complex expression is like a cluttered mess where you spend more time searching than actually building. So, by mastering the art of simplification, we're not just solving math problems; we're sharpening a skill that will help us in many areas of life. Now, let's roll up our sleeves and get to the fun part – the actual simplification process!
Step-by-Step Simplification
Okay, let's break this down. The first thing we need to do is expand those cubic binomials. Remember the formulas?
(a + b)³ = a³ + 3a²b + 3ab² + b³
(a − b)³ = a³ − 3a²b + 3ab² − b³
These formulas are our secret weapons here. They allow us to transform the cubic expressions into something we can actually work with. If you're not familiar with these formulas, it's worth taking a moment to memorize them or at least understand how they're derived (hint: the binomial theorem is your friend!). Now, let's substitute these expansions back into our original expression. We get:
(a³ + 3a²b + 3ab² + b³) − (a³ − 3a²b + 3ab² − b³) − 5a²b
Notice how we've replaced the cubic terms with their expanded forms. This is a crucial step because it turns the problem from dealing with exponents into dealing with addition and subtraction of terms. But we're not done yet! We still have that pesky subtraction in the middle. Remember, subtracting an expression is the same as adding the negative of that expression. So, let's distribute that negative sign through the second set of parentheses. This is a common spot for errors, so we'll take our time and be super careful with our signs. After distributing the negative sign, our expression looks like this:
a³ + 3a²b + 3ab² + b³ − a³ + 3a²b − 3ab² + b³ − 5a²b
See how the signs inside the parentheses have flipped? This is exactly what we want. Now, we're ready for the next step: combining like terms. This is where we gather all the terms that have the same variables raised to the same powers and add or subtract their coefficients. It's like sorting your socks – you put all the matching pairs together. So, let's identify our like terms. We have a³ terms, a²b terms, ab² terms, and b³ terms. We'll go through each type and combine them. This is where the magic happens, and our expression starts to simplify beautifully. Stay tuned as we unravel this algebraic knot!
Combining Like Terms: The Grand Finale
Alright, let's get down to the nitty-gritty and combine those like terms. Remember, we have:
a³ + 3a²b + 3ab² + b³ − a³ + 3a²b − 3ab² + b³ − 5a²b
First up, we have a³ terms. We've got a³ and -a³. These guys cancel each other out! It's like they're playing tug-of-war and the rope breaks right in the middle. They just disappear. So, we can cross them out. Next, let's tackle the a²b terms. We have 3a²b, 3a²b, and -5a²b. If we combine these, we get 3 + 3 - 5 = 1. So, we're left with 1a²b, which we can simply write as a²b. See how things are simplifying? It's like watching a complex puzzle come together piece by piece. Now, let's move on to the ab² terms. We have 3ab² and -3ab². Just like our a³ terms, these guys cancel out too! They're like perfect opposites, always balancing each other out. This leaves us with no ab² terms in our simplified expression. Finally, let's look at the b³ terms. We have b³ and b³. These are easy to combine: 1 + 1 = 2. So, we have 2b³. Now, let's put it all together. After combining all the like terms, our expression has transformed into:
2b³ + a²b
Whoa! Look how much simpler that is compared to where we started. It's like we've taken a messy room and organized it into a neat, tidy space. But hold on a second! We're not quite done yet. We need to compare our simplified expression to the given alternatives to find the correct answer. So, let's take a peek at those options and see if we've nailed it!
Comparing and Concluding
Okay, so we've simplified our expression to 2b³ + a²b. Now, let's look at the alternatives provided:
A) 6ab²
B) 2a³ + 6ab²
C) 4a²b
D) 8ab
Hmm, none of these seem to directly match our simplified expression, 2b³ + a²b. This might seem a bit puzzling at first, but let's not panic! Sometimes, the answer might be hidden in plain sight, or there might be a small trick involved. We need to double-check our work to make sure we haven't made any mistakes along the way. It's like being a detective and looking for clues. We'll go back through each step of our simplification process, from expanding the binomials to combining like terms, and make sure everything is spot on. It's a good habit to develop in math – always double-check your work! A small mistake can sometimes lead to a completely different answer. So, let's put on our detective hats and retrace our steps. We'll look closely at each step and make sure we haven't missed anything. This is where attention to detail really pays off. Sometimes, the answer is right in front of us, but we just need to look at it from a slightly different angle. So, let's get to it and make sure we've cracked this case!
Spotting the Error and Correcting Our Course
Alright, team, let's put on our detective hats and revisit our steps. We expanded (a + b)³ and (a - b)³, combined like terms, and ended up with 2b³ + a²b. But something's not quite right, as our answer doesn't match any of the options. This is a classic moment in problem-solving – time to retrace our steps and find the hidden culprit, a.k.a. the mistake! Let's rewind to the crucial step of expanding and simplifying:
(a + b)³ − (a − b)³ − 5a²b = (a³ + 3a²b + 3ab² + b³) − (a³ − 3a²b + 3ab² − b³) − 5a²b
(a³ + 3a²b + 3ab² + b³) − a³ + 3a²b − 3ab² + b³ − 5a²b
Now, let’s meticulously combine like terms: The a³ terms cancel out (a³ - a³ = 0). For the a²b terms, we have 3a²b + 3a²b − 5a²b = a²b. For the ab² terms, 3ab² - 3ab² also cancel out. And for the b³ terms, we have b³ + b³ = 2b³. So, after carefully combining them, we have:
6a²b + 2b³ - 5a²b
Oops! It seems we made a slight miscalculation when combining the like terms. Let's correct that now. Looking closely, the a²b terms combine to 3a²b + 3a²b - 5a²b = a²b. And the b³ terms combine to b³ + b³ = 2b³. It's easy to see how a small slip can happen, but the important thing is to catch it and learn from it. So, let's make the adjustment. Our simplified expression should be:
a²b + 2b³
But wait! Even with this correction, our expression still doesn't directly match any of the provided alternatives. This indicates that we need to carefully revisit the entire simplification process, paying close attention to every sign and operation. It's like being a detective piecing together clues – each step must be verified to ensure the solution is accurate. Sometimes, a fresh perspective or a second pair of eyes can help spot an oversight. Let's take a deep breath and meticulously review our work once more to uncover the correct path to the solution.
The Aha! Moment: Spotting the Real Culprit
Okay, detectives, let's dive back into our algebraic mystery. We've expanded, combined, and even double-checked, but our expression a²b + 2b³ still isn't waving hello from the options list. Time to channel our inner Sherlock Holmes and look for what we might have overlooked. We're going to go back to the very beginning and meticulously check each step, because sometimes, the trickiest errors hide in the most obvious places. Let's start with our original expression:
(a + b)³ − (a − b)³ − 5a²b
And our expansions:
(a + b)³ = a³ + 3a²b + 3ab² + b³
(a − b)³ = a³ − 3a²b + 3ab² − b³
These look good, so let's substitute them back in:
(a³ + 3a²b + 3ab² + b³) − (a³ − 3a²b + 3ab² − b³) − 5a²b
Now, the crucial part – distributing that negative sign. Let's be extra careful here:
a³ + 3a²b + 3ab² + b³ − a³ + 3a²b − 3ab² + b³ − 5a²b
Everything still seems in order. Now, let's combine those like terms again, super slowly and deliberately:
The a³ terms: a³ - a³ = 0. Perfect.
The a²b terms: 3a²b + 3a²b - 5a²b. Ah ha! Here's where the magic (or rather, the mistake) happened. We calculated this as a²b, but it should be 3 + 3 - 5 = 1, so it's indeed a²b. But let’s make sure we are not missing anything else.
The ab² terms: 3ab² - 3ab² = 0. Great.
The b³ terms: b³ + b³ = 2b³. Looking good.
So, we're back to a²b + 2b³. Still no match in our options. This is like a plot twist in our math mystery! What if the answer isn't in the simplification itself, but in how we interpret the options? Let’s step back and look at the big picture. We've meticulously checked our algebra, and we're confident in our simplified expression. Maybe the options are presented in a way that requires a little algebraic thinking outside the box. Let's revisit those options with a fresh perspective and see if we can crack the code.
Cracking the Code: The Correct Answer Revealed
Okay, fellow math detectives, let's put on our thinking caps one more time. We've simplified the expression to a²b + 2b³, and we've double, triple, and quadruple-checked our work. Yet, none of the options A) 6ab², B) 2a³ + 6ab², C) 4a²b, and D) 8ab seem to match. This is where we need to think outside the box and consider if there's another layer to this problem. Is there a common factor we can pull out? Can we rearrange terms to see if it clicks with any of the options? Sometimes, the answer is disguised in a slightly different form, and it's our job to unveil it. Let’s try factoring. Looking at a²b + 2b³, we can see that b is a common factor. Factoring out a b, we get:
b(a² + 2b²)
This is an interesting twist, but still not quite there in terms of matching our options. What if we made a mistake way back when? It's time to zoom out and look at the bigger picture. We’ve been so focused on the mechanics of simplification that we might have missed a forest-for-the-trees kind of situation. Remember, the key to solving complex problems is often not just about applying formulas, but about truly understanding the underlying concepts. Let's go back to the original problem:
(a + b)³ − (a − b)³ − 5a²b
And our expansions:
(a + b)³ = a³ + 3a²b + 3ab² + b³
(a − b)³ = a³ − 3a²b + 3ab² − b³
What if, instead of expanding everything out, we focused on the difference of cubes first? There's a handy formula for that:
A³ - B³ = (A - B)(A² + AB + B²)
But does this apply to binomial cubes? Let's try a different tack. Let's go back to our expanded form before we combined like terms. This is like going back to the crime scene to look for new evidence. We had:
a³ + 3a²b + 3ab² + b³ − a³ + 3a²b − 3ab² + b³ − 5a²b
Now, let's group the terms carefully:
(a³ - a³) + (3a²b + 3a²b - 5a²b) + (3ab² - 3ab²) + (b³ + b³)
This gives us:
0 + a²b + 0 + 2b³
So, we're back to a²b + 2b³. It seems like we're stuck in a loop! But wait a minute... what if we made a mistake copying the original problem? It might sound crazy, but sometimes the simplest explanation is the correct one. Let’s imagine, for a moment, that the problem was slightly different. What if the last term was different? This is a long shot, but let’s entertain the possibility. If we can’t find an error in our work, maybe the error is in the problem itself. So, let’s take a break from the algebra for a second and consider the bigger picture. What are the common types of algebraic simplifications? What kind of answers are we expecting? And are we sure we've copied the problem correctly? It’s time to zoom out and look at the whole landscape of the problem, not just the individual trees.
The Solution!
Guys, after a thorough review of the steps and a bit of algebraic gymnastics, it seems there was a mistake in the original problem statement or the provided alternatives. The correct simplified expression, based on the given expression (a + b)³ − (a − b)³ − 5a²b, is a²b + 2b³. However, none of the options (A) 6ab², (B) 2a³ + 6ab², (C) 4a²b, and (D) 8ab match this result. This situation highlights the importance of not only mastering the simplification techniques but also critically evaluating the problem itself and the potential for errors in the given information. Sometimes, the most challenging part of a math problem isn't the math itself, but recognizing when something isn't quite right. In conclusion, while we've successfully simplified the expression, the correct answer isn't among the provided choices. It might be worth double-checking the original problem or the alternatives for any typos or errors. Keep up the great work, and remember, problem-solving is just as much about critical thinking as it is about calculations!
