Expanding & Simplifying: (-4b - 3b^2 - 4)(2b - 4)

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Hey guys! Let's dive into expanding and simplifying the algebraic expression (-4b - 3b^2 - 4)(2b - 4). This type of problem often appears in algebra, and mastering it is super useful for tackling more complex equations later on. We'll break it down step by step, making it easy to follow along. So, grab your pencils and let's get started!

Understanding the Distributive Property

At the heart of expanding expressions like this is the distributive property. Think of it like this: you're distributing one term across all the terms within another set of parentheses. Basically, every term in the first set of parentheses needs to multiply every term in the second set. It might sound a little intimidating, but it's really just a matter of careful multiplication and organization.

To kick things off, let's identify our terms. In the first set of parentheses, we have -4b, -3b^2, and -4. In the second set, we have 2b and -4. This means we'll have a total of 3 * 2 = 6 multiplications to perform. Organization is key here, so let's take it one step at a time to avoid any silly mistakes. Trust me, we've all been there!

Before we jump into the actual multiplication, let’s quickly recap the basic rules of multiplying terms with variables and exponents. Remember, when you multiply terms with the same base (like 'b' in our case), you add their exponents. For example, b * b is b^2, and b^2 * b is b^3. Also, don't forget the signs! A negative times a positive is a negative, and a negative times a negative is a positive. These little details can make a big difference in the final answer. Keeping these rules in mind will help us stay on track and ensure we get the correct result. Now, let's get to the fun part – the multiplication!

Step-by-Step Expansion

Now, let's get into the nitty-gritty of the expansion. We'll go through each multiplication one by one, making sure we keep track of our signs and exponents. Here’s how it breaks down:

  1. Multiply -4b by both terms in the second parentheses:

    • -4b * 2b = -8b^2
    • -4b * -4 = 16b

  2. Multiply -3b^2 by both terms in the second parentheses:

    • -3b^2 * 2b = -6b^3
    • -3b^2 * -4 = 12b^2

  3. Multiply -4 by both terms in the second parentheses:

    • -4 * 2b = -8b
    • -4 * -4 = 16

See? It’s just a series of smaller multiplications. Now we have all the pieces of the puzzle. Our expanded expression looks like this: -8b^2 + 16b - 6b^3 + 12b^2 - 8b + 16. We're not done yet, though. The next step is to simplify this by combining like terms. This is where things start to clean up and the expression becomes more manageable. So, let's move on to the next stage and bring this all together!

Combining Like Terms

The next crucial step in simplifying our expression is to combine like terms. What are like terms, you ask? They're simply terms that have the same variable raised to the same power. For instance, b^2 terms can only be combined with other b^2 terms, and b terms can only be combined with other b terms. Constants (the numbers without any variables) can only be combined with other constants.

Looking back at our expanded expression, -8b^2 + 16b - 6b^3 + 12b^2 - 8b + 16, we can identify the like terms:

  • b^3 terms: We only have one term with b^3, which is -6b^3.
  • b^2 terms: We have -8b^2 and 12b^2. Combining these gives us -8b^2 + 12b^2 = 4b^2.
  • b terms: We have 16b and -8b. Combining these gives us 16b - 8b = 8b.
  • Constants: We only have one constant, which is 16.

Now, let's rewrite our expression with the like terms combined. This gives us -6b^3 + 4b^2 + 8b + 16. See how much simpler it looks? This is the simplified form of our expanded expression. We've taken a longer, more complex expression and condensed it into something much cleaner and easier to work with. This process of combining like terms is a fundamental skill in algebra, and it's essential for solving equations and simplifying more complex problems. Next, we'll talk about putting this in standard form, just to give our answer that final polished look!

Writing in Standard Form

To give our final answer a polished and professional look, we should write it in standard form. Standard form simply means arranging the terms in descending order of their exponents. In other words, we start with the term that has the highest power and work our way down to the constant term.

Looking at our simplified expression, -6b^3 + 4b^2 + 8b + 16, we can easily rearrange it into standard form. The term with the highest power is -6b^3, followed by 4b^2, then 8b, and finally the constant 16. So, in standard form, our expression becomes -6b^3 + 4b^2 + 8b + 16. Notice that the order of the terms has changed, but the expression's value remains the same.

Writing expressions in standard form isn't just about aesthetics; it also makes it easier to compare and combine expressions. When everything is in a consistent format, it's much simpler to identify like terms and perform operations like addition and subtraction. Plus, it's a good habit to develop because many algebraic concepts and theorems are based on expressions being in standard form. So, always remember to put your final answers in standard form to ensure clarity and accuracy. Now, let's recap what we've done and highlight the key takeaways from this problem.

Final Result and Conclusion

Alright guys, let's wrap things up! We started with the expression (-4b - 3b^2 - 4)(2b - 4) and went through a step-by-step process to expand and simplify it. We used the distributive property to multiply each term in the first parentheses by each term in the second parentheses. This gave us a longer expression, which we then simplified by combining like terms. Finally, we arranged our simplified expression in standard form, resulting in our final answer: -6b^3 + 4b^2 + 8b + 16.

The key takeaways from this problem are:

  • Distributive Property: Remember to multiply each term in one set of parentheses by every term in the other set.
  • Combining Like Terms: Only combine terms that have the same variable raised to the same power.
  • Standard Form: Write your final answer in descending order of exponents for clarity and consistency.

By mastering these steps, you'll be well-equipped to tackle similar algebraic expressions with confidence. Keep practicing, and you'll become a pro at expanding and simplifying in no time! Remember, algebra is all about breaking down complex problems into smaller, manageable steps. So, keep a cool head, stay organized, and you'll nail it every time. Great job, and keep up the awesome work!