Multiplying Polynomials: A Step-by-Step Guide
Hey everyone! Today, we're diving into the world of polynomial multiplication. Specifically, we're going to break down how to find the product of two polynomials: and . Don't worry if it sounds a little intimidating; we'll walk through it step by step, making sure you grasp every detail. This is a fundamental concept in algebra, and once you get the hang of it, you'll be able to tackle more complex problems with ease. So, grab your pencils and let's get started!
This kind of problem is super common in algebra, and it's all about applying the distributive property. Remember that property? It's the one that lets us multiply each term in one set of parentheses by each term in the other set. We'll break down the multiplication into smaller, manageable chunks. Think of it like this: We're going to distribute the 4z from the first set of parentheses across all the terms in the second set, and then we'll do the same for the -3. By breaking it down this way, we can avoid making mistakes and keep our work organized. It's like a recipe; following each step carefully will get us the right answer every time. In this case, our recipe is the distributive property, and our ingredients are the terms within the polynomials. Let's start the breakdown.
Now, let's look at the first term. We have . Let's start by multiplying 4z by each term in the second set of parentheses. First, we'll multiply 4z by -z². When multiplying terms with variables, we multiply the coefficients (the numbers in front of the variables) and add the exponents of the variables. In this case, 4 times -1 (the coefficient of z²) is -4, and z times z² is z³. So, our first product is -4z³. Next, we'll multiply 4z by 2z. Four times two is eight, and z times z is z². Thus, our second product is 8z². Finally, we multiply 4z by -4. Four times -4 is -16, and the z remains. This gives us -16z. So, when we distribute 4z, we get -4z³ + 8z² - 16z. We've conquered the first part! Now, we are halfway. So far so good, right?
Next, we'll focus on the second term in the first set of parentheses, which is -3. We'll multiply this by each term in the second set of parentheses, just like we did with the 4z. First, we multiply -3 by -z². A negative times a negative is a positive, so -3 times -1 (the coefficient of z²) is 3, and we have z². Thus, our first product is 3z². Then, we multiply -3 by 2z. Negative three times two is -6, and the z remains, giving us -6z. Finally, we multiply -3 by -4. A negative times a negative is a positive, so -3 times -4 is 12. This gives us +12. Therefore, when we distribute -3, we get 3z² - 6z + 12. We're getting closer to our final answer. Now, we've multiplied out the entire expression. It is like the second part of our recipe, and we are almost done!
Combining Like Terms and Simplifying
Alright, guys, we've done the multiplication part. Now comes the simplifying part, where we combine like terms. Remember, like terms are terms that have the same variable raised to the same power. In our expanded expression, we need to look for terms that we can add or subtract. Our expanded form is -4z³ + 8z² - 16z + 3z² - 6z + 12. Let's identify the like terms. We have one z³ term: -4z³. Then, we have two z² terms: 8z² and 3z². We also have two z terms: -16z and -6z. And finally, we have a constant term: +12.
To combine like terms, we add or subtract their coefficients. So, let's start with the z² terms: 8z² + 3z² equals 11z². Next, we combine the z terms: -16z - 6z equals -22z. The -4z³ term stays as it is, as there are no other z³ terms to combine it with. And the constant term, +12, also stays as is, because there are no other constant terms. After combining all of our like terms, we get -4z³ + 11z² - 22z + 12. This is our simplified answer. We've successfully multiplied the polynomials and simplified the result! Congratulations, you made it through! Now let's put it all together.
So, starting with the original expression: . We first distributed the 4z, resulting in -4z³ + 8z² - 16z. Then, we distributed the -3, giving us 3z² - 6z + 12. Then, we combine like terms. Combining the z² terms (8z² + 3z²) gives us 11z². Combining the z terms (-16z - 6z) equals -22z. The -4z³ term stays the same, and the constant term, +12, also stays the same. Thus, the final simplified product is -4z³ + 11z² - 22z + 12. That's the final answer! You see, it wasn't as hard as it looked at the start, right? You just need to follow the steps carefully and keep track of your signs. The main thing is to stay organized and patient. Before you know it, you'll be a pro at multiplying polynomials.
Tips and Tricks for Success
To really ace these kinds of problems, here are a few tips and tricks to keep in mind. First of all, always double-check your work. It's easy to make small mistakes with the signs or when multiplying the coefficients, so take a moment to review each step. Secondly, stay organized. Write out each step clearly and keep your work neat. This will help you avoid errors and make it easier to find any mistakes if they do occur. Also, remember to pay close attention to the signs (positive and negative). It's a very common place to make a mistake. A negative times a negative is positive, and a positive times a negative is negative. Do not forget that. Make sure to combine like terms correctly. Adding or subtracting the coefficients of the variables correctly will allow you to get the correct answer. The distributive property is your friend; use it! It's the core concept behind this type of problem, so make sure you understand it well. Practice makes perfect. The more problems you solve, the more comfortable and confident you'll become. So, keep practicing! Do lots of examples.
Finally, remember that math, like any skill, improves with practice. The more you work through these problems, the more familiar you'll become with the process. Don't be discouraged if it seems tough at first. Keep practicing, reviewing the steps, and you'll become a pro in no time! So, keep practicing. This is such a great way to grow your knowledge. Also, remember there are a ton of resources. There are many great online resources like Khan Academy, YouTube videos, and even your textbook can provide lots of practice problems and explanations. Take advantage of them! You've got this!
Conclusion
So, there you have it! We've successfully multiplied and simplified the polynomials . Remember the key steps: distribute each term, combine like terms, and double-check your work. This is a fundamental concept in algebra, so congratulations on mastering it! Keep practicing, and you'll be well on your way to success in your math journey! Keep up the great work, everyone! Keep practicing, and keep learning! You are doing great.