Expand (3x + 5y)(2x + 7y + 9z): A Simple Guide

Hey guys! Today, we're diving into expanding the expression (3x + 5y)(2x + 7y + 9z). This might look a bit intimidating at first, but don't worry, we'll break it down step-by-step. Think of it like a puzzle – each piece fits perfectly if you know where to put it. So, grab your pencils and let’s get started!

Understanding the Basics of Expansion

When we talk about expanding expressions in mathematics, we essentially mean getting rid of the parentheses by multiplying each term inside one set of parentheses by each term inside the other set. It's like distributing a package to everyone in a room – each person gets a piece! The core principle here is the distributive property, which states that a(b + c) = ab + ac. We’re just going to apply this on a slightly larger scale.

Before we jump into our main problem, let's quickly revisit a simpler example to make sure we're all on the same page. Suppose we want to expand (a + b)(c + d). What do we do? We take the first term in the first parenthesis, ‘a’, and multiply it by both terms in the second parenthesis, ‘c’ and ‘d’. That gives us ac + ad. Then, we take the second term in the first parenthesis, ‘b’, and do the same thing – multiply it by ‘c’ and ‘d’, resulting in bc + bd. Finally, we add all these terms together: ac + ad + bc + bd. This systematic approach is what we'll use for our more complex problem as well. Remembering this foundational method ensures we don’t miss any terms and keep our expansion accurate. Think of it as building a house; a strong foundation is key for a sturdy structure. Similarly, a solid grasp of basic expansion techniques is crucial for tackling more advanced algebraic problems.

Step-by-Step Expansion of (3x + 5y)(2x + 7y + 9z)

Okay, let's tackle the main event: expanding (3x + 5y)(2x + 7y + 9z). The trick here is to take it one step at a time. We'll use the same distributive property we just discussed, but this time, we have three terms in the second parenthesis instead of two. No sweat! We can handle this.

First, let's take the first term from the first parenthesis, which is 3x. We're going to multiply this by each term in the second parenthesis: 2x, 7y, and 9z. So, we get:

• 3x * 2x = 6x²
• 3x * 7y = 21xy
• 3x * 9z = 27xz

Now, let’s move on to the second term in the first parenthesis, which is 5y. We’ll do the same thing – multiply 5y by each term in the second parenthesis:

• 5y * 2x = 10xy
• 5y * 7y = 35y²
• 5y * 9z = 45yz

Notice how we're systematically working through each term. This is key to avoiding mistakes and keeping things organized. Now we have all the individual products. What’s next? We need to add them all together! This is where we combine like terms, which we'll cover in the next section.

Combining Like Terms

After expanding, we usually end up with a bunch of terms. The next step is to simplify things by combining like terms. Like terms are those that have the same variables raised to the same powers. Think of it as sorting socks – you pair up the ones that match! In our expanded expression, we have terms like x², xy, xz, y², and yz. We need to identify which of these are “like” each other.

Looking at our expanded form, which is currently:

6x² + 21xy + 27xz + 10xy + 35y² + 45yz

We can see that we have two terms with ‘xy’: 21xy and 10xy. These are like terms, so we can combine them. All the other terms (6x², 27xz, 35y², and 45yz) don’t have any other terms that match them, so they'll stay as they are.

To combine 21xy and 10xy, we simply add their coefficients (the numbers in front of the variables). So, 21xy + 10xy = 31xy. This is just like saying 21 apples + 10 apples = 31 apples! It's a straightforward arithmetic operation on the coefficients.

After combining these like terms, our expression looks much cleaner and simpler. This step is crucial for getting to the most simplified form of our expanded expression. Simplifying makes it easier to work with the expression in future calculations or problem-solving scenarios. It’s like decluttering your workspace – a clean space makes it easier to focus and be productive. In math, a simplified expression makes it easier to see patterns, solve equations, and understand the relationships between variables.

The Final Simplified Expression

Alright, guys, we've done the hard work! We expanded the expression and combined the like terms. Now, let's put it all together to get our final answer. Remember, we started with:

(3x + 5y)(2x + 7y + 9z)

We expanded it to:

6x² + 21xy + 27xz + 10xy + 35y² + 45yz

And then we combined like terms (21xy and 10xy) to get 31xy. So, our final simplified expression is:

6x² + 31xy + 27xz + 35y² + 45yz

Isn't that satisfying? We took a somewhat complex expression and broke it down into a much simpler form. This is the power of algebraic manipulation! You can see how each step contributes to the final result, and how organizing your work makes the process much smoother. We've gone from a product of two binomials to a clear, expanded polynomial. This skill is super important in algebra and beyond, as it forms the basis for solving equations, graphing functions, and tackling more advanced mathematical concepts. Think of it as mastering the fundamentals of a sport – once you have those down, you can take on more challenging techniques and strategies. Similarly, a solid understanding of expansion and simplification will set you up for success in your mathematical journey. Good job, everyone; we nailed it!

Tips for Avoiding Mistakes

Expanding expressions can be a bit like navigating a maze – there are lots of turns and it’s easy to make a wrong step. But don't worry, I've got some tips and tricks to help you avoid common mistakes and stay on the right path. Think of these as your map and compass for the algebraic wilderness!

1. Double-Check Your Signs: This is a big one. It’s super easy to lose track of a negative sign when you’re multiplying. Always, always double-check that you’ve applied the correct sign to each term. A small mistake with a sign can throw off your entire answer. Make it a habit to circle or highlight negative signs as you work through the problem – it's like putting a flag on a potential danger zone.

2. Stay Organized: Write neatly and keep your work organized. This will make it much easier to spot mistakes and follow your own steps later on. When you’re dealing with multiple terms, a well-organized workspace is your best friend. Consider writing each step on a new line, or using a grid-like layout to keep your terms aligned. This isn't just about aesthetics; it's about making your work as clear as possible so you can review it effectively.

3. Take It One Step at a Time: Don't try to rush through the expansion. Break it down into smaller steps and tackle each one individually. Remember, we walked through this problem term by term – that’s the key to accuracy. Trying to do too much at once can lead to errors, so pace yourself and focus on getting each step right.

4. Combine Like Terms Carefully: Make sure you’re only combining terms that are actually “like.” Remember, they need to have the same variables raised to the same powers. It’s like sorting your laundry – you wouldn't throw a red sock in with the whites, right? Similarly, you shouldn't combine x² terms with x terms. Double-check that the variables and their exponents match before you combine coefficients.

5. Practice, Practice, Practice: The more you practice expanding expressions, the better you’ll get at it. It's like learning to ride a bike – the first few times might be wobbly, but with practice, it becomes second nature. Work through plenty of examples, and don’t be afraid to make mistakes. Mistakes are learning opportunities! The more you practice, the more confident and accurate you'll become.

By keeping these tips in mind, you'll be well-equipped to tackle any expansion problem that comes your way. Think of it as building a toolkit – each tip is a tool that helps you navigate the challenges of algebra. So, keep practicing, stay organized, and remember to double-check your work. You’ve got this!

Conclusion

So, there you have it, guys! We've successfully expanded the expression (3x + 5y)(2x + 7y + 9z) step-by-step. We covered the basics of expansion, the distributive property, combining like terms, and some handy tips for avoiding mistakes. You've added another valuable tool to your algebraic toolbox!

Remember, expanding expressions is a fundamental skill in algebra, and mastering it will help you in so many areas of math. It's like learning the alphabet in language – you need it to form words, sentences, and ultimately, communicate effectively. In math, expansion allows you to simplify complex expressions, solve equations, and understand relationships between variables.

Keep practicing, keep exploring, and most importantly, keep enjoying the process of learning. Math can be challenging, but it's also incredibly rewarding. Every time you solve a problem, you're building your problem-solving skills and your confidence. Think of each problem as a puzzle, and you're becoming a master puzzle-solver. So, go forth and expand those expressions with confidence! You’ve got this!