Expanding Polynomials: A Step-by-Step Guide

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Hey guys! Today, we're diving into the world of algebra and tackling a problem that might seem a bit intimidating at first: expanding and simplifying the expression (3a - 2b)(9a² + 6ab + 4b²). Don't worry, it's not as scary as it looks! We'll break it down into easy-to-follow steps, making sure you understand every bit of the process. This is a fundamental skill in algebra, and mastering it will help you with more complex problems down the road. We will use several key concepts like the distributive property, combining like terms and understanding the rules of exponents to accomplish our goals. In this article we'll not only find the solution but explain the 'why' behind each step, making sure you are building a solid foundation in algebra. We'll explore how to write this expression in standard form, which means arranging the terms in descending order of their degree. This process helps us organize our polynomials and makes them easier to analyze and work with.

Let's start with the distributive property. This property is the cornerstone of expanding expressions like this. It tells us that we need to multiply each term in the first set of parentheses by each term in the second set of parentheses. Think of it like this: each term in the first part gets to 'visit' and multiply with each term in the second part. The goal is to make sure every possible multiplication happens once, and we don't miss any terms. This process requires careful attention to detail, but we will get through it together.

Remember to pay close attention to the signs (+ and -) because a tiny mistake here can change the entire outcome. Keeping track of these signs is super important to ensure that we correctly combine the terms. We will also need to be familiar with the concept of 'like terms'. Like terms have the same variables raised to the same powers. For instance, 3ab and -5ab are like terms because they both have a and b raised to the power of 1. We can add or subtract like terms to simplify the expression. So let's get started.

Breaking Down the Expression

Okay, let's get down to business. Our expression is (3a - 2b)(9a² + 6ab + 4b²). The first step is to multiply each term in the first parenthesis by each term in the second parenthesis. This means we'll multiply 3a by each term in the second set and then multiply -2b by each term in the second set. Let's do it step by step:

First, multiply 3a by the second set:

  • 3a * 9a² = 27a³
  • 3a * 6ab = 18a²b
  • 3a * 4b² = 12ab²

Next, multiply -2b by the second set:

  • -2b * 9a² = -18a²b
  • -2b * 6ab = -12ab²
  • -2b * 4b² = -8b³

See? Not so bad, right? We've successfully distributed the terms, and now we have a collection of terms that we need to simplify. Each multiplication step results in a new term, and we must now collect these terms to complete our task. We did a careful process to make sure we did not miss any terms or make any mistakes with the signs. We kept track of the variables and their exponents. Now, we just need to put it all together and tidy things up a bit by combining all of the like terms. This expansion step is crucial for simplifying the expression and preparing it for the next phase: combining like terms.

Combining Like Terms

Now that we've expanded the expression, we have a bunch of terms. Our next goal is to combine any like terms. Remember, like terms have the same variables raised to the same powers. Look closely at the terms we got from the expansion: 27a³, 18a²b, 12ab², -18a²b, -12ab², and -8b³. Let's see if we can find any pairs of like terms.

In this case, we can find a pair of like terms: 18a²b and -18a²b. When we add these together, they cancel each other out (18 - 18 = 0). We also have 12ab² and -12ab², which also cancel each other out. The remaining terms are 27a³ and -8b³. So when we combine all of the like terms, the expression gets significantly simplified. The step of combining like terms helps us reduce the complexity of the expression and makes it easier to handle and understand.

By carefully comparing the terms, we identified which ones were identical in terms of their variables and powers and added or subtracted the coefficients accordingly. The whole process is about making our expression as simple and understandable as possible. The skill of combining like terms is a vital part of algebra and will be used as we start solving different types of equations. Simplifying expressions is an important skill, as it reduces complexity and allows you to manipulate them more easily in later stages of a problem, like finding values or graphing equations. Understanding and mastering these steps is a key part of solving algebraic problems and moving forward in mathematics.

Writing in Standard Form

Great job, guys! After combining the like terms, our expression becomes 27a³ - 8b³. But we are not quite done yet! The last step is to write the result in standard form. Standard form for a polynomial means arranging the terms in descending order of their degree (the sum of the exponents of the variables in each term). In our case, we have two terms, 27a³ and -8b³. The term 27a³ has a degree of 3 (because the exponent of a is 3), and the term -8b³ has a degree of 3 (because the exponent of b is 3). Since both have the same degree, we write the terms in descending order based on the variable we choose to prioritize. In this case, both variables have the same power, so our expression is already in standard form.

So, the final answer in standard form is 27a³ - 8b³. Congrats, you did it! You successfully expanded the expression and wrote it in standard form. By following these steps, you've not only solved the problem but also reinforced your understanding of important algebraic concepts. Knowing how to expand and simplify polynomial expressions is a great skill to have.

Throughout the process, we focused on the importance of each step, including the correct application of properties, careful distribution, and combining like terms. Mastering these methods will give you a solid foundation in algebra. Understanding the rules of exponents and how to apply them is a crucial part of simplifying polynomials. By mastering these techniques, you'll be able to confidently tackle more complex mathematical challenges. With this new understanding, you should be ready to apply these skills to tackle more complex problems. Keep practicing, and you will become a pro at expanding and simplifying polynomials!

Summary and Key Takeaways

Let's quickly recap what we did:

  1. Expanded the expression using the distributive property. Remember to multiply each term in the first parenthesis by each term in the second parenthesis.
  2. Combined like terms. Identify the terms with the same variables and exponents and simplify them.
  3. Wrote the final answer in standard form by arranging the terms in descending order of their degree.

I hope this detailed guide has helped you better understand how to expand and simplify polynomial expressions. Keep practicing, and you will master these skills in no time. If you have questions, don't hesitate to ask! Practice makes perfect, and the more you work with these concepts, the more comfortable you will become. Keep up the fantastic work!