Simplifying Expressions A Step-by-Step Guide To -2b(-3b^2+4a-5c)
Hey guys! Let's dive into simplifying algebraic expressions. Today, we're tackling a specific problem: -2b(-3b^2+4a-5c). Don't worry, it might look intimidating at first, but we'll break it down step by step. Our main goal here is to understand how to distribute and combine like terms to make complex expressions simpler and easier to work with. Mastering these skills is super important for algebra and beyond, so let's get started!
Understanding the Problem: -2b(-3b^2+4a-5c)
Before we jump into solving, let's make sure we fully grasp what the problem is asking. We have an algebraic expression that involves variables (a, b, and c) and coefficients (the numbers in front of the variables). The core of this problem is the distributive property. Remember that? It's when you multiply a term outside the parentheses by each term inside the parentheses. The expression -2b(-3b^2+4a-5c) means we need to multiply -2b by each of the three terms inside the parentheses: -3b^2, 4a, and -5c. Think of it like this: -2b is trying to share itself with everyone inside the parenthesis party! We need to make sure everyone gets their fair share.
Keywords like “simplify,” “expression,” and the actual terms like “-2b,” “-3b^2,” “4a,” and “-5c” are crucial here. They tell us exactly what kind of problem we're dealing with. The presence of parentheses is a big clue that we’ll be using the distributive property. Understanding the individual components – the coefficients, variables, and signs – is essential to getting the correct answer. We also need to keep an eye on those negative signs! They can be tricky, but we’ll handle them with care.
Now, why is this important? Well, simplified expressions are much easier to work with. Imagine trying to solve a complex equation with this expression still in its original form. It would be a mess! By simplifying, we reduce the chances of making mistakes and make the entire process smoother. So, let's get this party started and distribute that -2b!
Step-by-Step Solution: Distributing -2b
Okay, let's get our hands dirty and start simplifying! The first step, as we discussed, is to use the distributive property. We need to multiply -2b by each term inside the parentheses. So, we have three separate multiplications to perform:
- -2b * -3b^2
- -2b * 4a
- -2b * -5c
Let's tackle each one individually. For the first one, -2b * -3b^2, remember that when you multiply terms with the same base (in this case, 'b'), you add their exponents. So, b * b^2 becomes b^(1+2) = b^3. Also, a negative times a negative is a positive! So, -2 * -3 = 6. Putting it all together, -2b * -3b^2 = 6b^3. See? Not so scary!
Next up, -2b * 4a. This one is a bit simpler in terms of exponents. We just multiply the coefficients (-2 * 4 = -8) and keep the variables 'b' and 'a'. So, -2b * 4a = -8ab. Remember to keep the order alphabetical; it’s a good habit to get into.
Finally, we have -2b * -5c. Again, a negative times a negative is a positive. -2 * -5 = 10. We also have the variables 'b' and 'c', so we simply write them next to each other. -2b * -5c = 10bc. Awesome!
Now, let's put these three results together. We've distributed -2b across all the terms inside the parentheses. We now have a new, expanded expression. This is a huge step forward, and you're doing great!
Combining Like Terms: The Next Step
So, after distributing, we have this expression: 6b^3 - 8ab + 10bc. Now, the next crucial step in simplifying algebraic expressions is to check for like terms. But what exactly are like terms? Like terms are terms that have the same variables raised to the same powers. They can have different coefficients (the numbers in front), but the variable part must match exactly.
For example, 3x^2 and -5x^2 are like terms because they both have the variable 'x' raised to the power of 2. However, 3x^2 and 3x are not like terms because the powers are different (2 versus 1). Similarly, 2xy and -4yx are like terms because they have the same variables (x and y) raised to the same powers (1 each), even though the order of the variables is different. Remember, multiplication is commutative, meaning the order doesn't matter (xy is the same as yx).
Now, let's go back to our expression: 6b^3 - 8ab + 10bc. Take a close look. Do you see any like terms? Nope! That’s right. We have a b^3 term, an ab term, and a bc term. None of these have the same variable combination. This means we can't combine anything further. Sometimes, you'll be able to combine like terms by adding or subtracting their coefficients, but in this case, we’re already at the simplest form. This is actually good news! It means we're one step closer to the final answer.
Recognizing like terms is a fundamental skill in algebra. It allows us to condense expressions and make them more manageable. Practice identifying like terms in various expressions, and you'll become a pro in no time!
The Final Simplified Expression
Alright, we've done the hard work! We distributed the -2b, and we checked for like terms. Guess what? We've reached the end of our journey! Our simplified expression is:
6b^3 - 8ab + 10bc
That's it! We took the original expression, -2b(-3b^2+4a-5c), and simplified it down to this neat and tidy form. Remember, this expression is equivalent to the original, but it's much easier to understand and work with. We’ve essentially cleaned up the messy version and presented a polished, simplified version.
Let's quickly recap what we did: First, we distributed the -2b across all the terms inside the parentheses. This involved multiplying -2b by -3b^2, 4a, and -5c. We had to pay close attention to the signs (negative times negative is positive!) and the exponents (remember to add them when multiplying terms with the same base). Then, we carefully checked for like terms. In this case, there weren't any, so we couldn't combine anything further. And that led us to our final answer.
This process of simplifying expressions is a cornerstone of algebra. It's used in countless problems, from solving equations to graphing functions. So, mastering this skill is super important. Practice with different expressions, and you'll become more confident and efficient. You’ll start seeing these problems not as daunting challenges, but as puzzles you can easily solve!
Real-World Applications and Why This Matters
Okay, so we've simplified this expression, but you might be thinking,
