Multiply (3x-6)(2x^2-7x+1): A Step-by-Step Guide

by Omar Yusuf 49 views

Introduction

Hey guys! Today, we're diving into a cool math problem where we need to find the product of two expressions: (3x - 6) and (2x^2 - 7x + 1). This kind of problem is super common in algebra, and it's all about using the distributive property (which might sound fancy, but it's actually pretty straightforward). We're going to break it down step by step, so don't worry if it seems a bit intimidating at first. By the end of this article, you'll be a pro at multiplying polynomials! Stick around, and let's get started on this mathematical adventure together. We'll make sure to keep it simple and fun, because who said math can't be enjoyable? Let's jump right into understanding how to tackle this problem.

Understanding Polynomial Multiplication

Before we jump into the specific problem, let's chat a bit about the general idea of polynomial multiplication. Think of it like this: you're not just multiplying numbers; you're multiplying expressions that have variables (like 'x') and constants (just regular numbers). The key here is the distributive property, which basically means each term in the first expression needs to be multiplied by each term in the second expression. It’s like saying everyone gets a turn to shake hands! For instance, if you have something like (a + b)(c + d), you need to do a * c, a * d, b * c, and b * d, then add them all up. This might sound like a mouthful, but it’s just about being systematic and making sure you don't miss any combinations. When you're working with polynomials that have more terms (like our problem today), this systematic approach is even more important. Trust me, once you get the hang of this, you'll see it's not as scary as it looks. Polynomial multiplication is a fundamental skill in algebra, and mastering it opens the door to solving more complex equations and problems. So, let’s keep this concept in mind as we move forward and tackle our main problem. We'll be applying this distributive property step-by-step, so you can see exactly how it works in action.

Step-by-Step Solution

Okay, let's get down to the nitty-gritty and solve (3x - 6)(2x^2 - 7x + 1) together! Remember, we're using the distributive property, so every term in the first expression needs to shake hands with every term in the second expression. First up, we'll take 3x and multiply it by each term in the second expression:

  • 3x * 2x^2 = 6x^3
  • 3x * -7x = -21x^2
  • 3x * 1 = 3x

Now, let's move on to the second term in our first expression, which is -6. We'll do the same thing, multiplying it by each term in the second expression:

  • -6 * 2x^2 = -12x^2
  • -6 * -7x = 42x
  • -6 * 1 = -6

Great! Now we have all the individual products. The next step is to add them all together. So, we have: 6x^3 - 21x^2 + 3x - 12x^2 + 42x - 6. But we're not done yet! We need to combine like terms – that means terms with the same power of x. Let’s gather them up:

  • x^3 terms: 6x^3 (there's only one)
  • x^2 terms: -21x^2 and -12x^2 (which combine to -33x^2)
  • x terms: 3x and 42x (which combine to 45x)
  • Constants: -6 (just one constant term)

Finally, we put it all together: 6x^3 - 33x^2 + 45x - 6. And that's our answer! See, it's all about breaking it down into manageable steps. We multiplied each term, added the results, and combined like terms. You’ve nailed it!

Detailed Breakdown of Each Step

Let’s really dive into the nitty-gritty and break down each step of multiplying (3x - 6)(2x^2 - 7x + 1). This is where we make sure every little detail is crystal clear. So, as we mentioned earlier, the cornerstone of this process is the distributive property. Think of it as a mathematical handshake – each term in the first expression has to “shake hands” (i.e., multiply) with every term in the second expression. We started with 3x from the first expression. This guy needs to multiply with every term in the second expression (2x^2, -7x, and 1). So, we have:

  • 3x * 2x^2. Remember when multiplying variables with exponents, you add the exponents. So, x times x squared is x cubed. Thus, 3x * 2x^2 equals 6x^3.
  • Next up, 3x * -7x. Here, we multiply the coefficients (3 and -7) and add the exponents of x (which are both 1). So, we get -21x^2.
  • Lastly, 3x * 1, which is just 3x. Easy peasy!

Now, we move onto the second term in the first expression: -6. This one also needs to multiply with every term in the second expression:

  • -6 * 2x^2. Multiply the coefficients: -6 times 2 is -12. So, we get -12x^2.
  • Then, -6 * -7x. A negative times a negative is a positive, so -6 times -7 is 42. That gives us 42x.
  • Finally, -6 * 1, which is simply -6.

So far, so good! We’ve done all the multiplications. Now comes the part where we add everything together. This is where it’s super important to be organized and keep track of all the terms. We line them up: 6x^3 - 21x^2 + 3x - 12x^2 + 42x - 6. But we're not quite done yet. We need to simplify by combining like terms. Like terms are those that have the same variable raised to the same power. It’s like grouping apples with apples and oranges with oranges. We look for terms with x^3, x^2, x, and the constants. In our case:

  • We have one x^3 term: 6x^3.
  • We have two x^2 terms: -21x^2 and -12x^2. When we combine these, we get -33x^2.
  • We have two x terms: 3x and 42x. Adding these gives us 45x.
  • And we have one constant term: -6.

Now, we put it all together in one beautiful, simplified expression: 6x^3 - 33x^2 + 45x - 6. And there you have it! We've walked through each step in detail, making sure you understand exactly how we arrived at the solution. Remember, the key is to take it slow, be organized, and don't rush. Polynomial multiplication might seem tricky at first, but with practice, you’ll become a pro in no time!

Common Mistakes to Avoid

Alright, let's chat about some common mistakes people often make when multiplying polynomials, especially in a problem like (3x - 6)(2x^2 - 7x + 1). Knowing these pitfalls can save you a lot of headaches and help you nail these problems every time. One of the biggest culprits is forgetting to distribute properly. Remember, every term in the first expression needs to multiply with every term in the second expression. It’s easy to miss one or two, especially when there are more terms involved. So, always double-check that you've hit every combination. Another frequent mistake is messing up the signs. A negative times a negative is a positive, a negative times a positive is a negative – these rules are crucial. Write down each step carefully, and pay close attention to the signs. It’s so easy to make a small sign error that throws off the entire answer. Exponents can also be tricky. When you multiply terms with exponents, you add the exponents, not multiply them. For example, x * x^2 is x^3, not x^2. Keep this rule in mind, and you'll avoid a common trap. Combining like terms is another area where mistakes can happen. Make sure you're only combining terms that have the same variable and the same exponent. You can't combine x^2 and x, for instance. It's like trying to add apples and oranges – they're just not the same. Organization is key here. Write out all the terms after you've done the multiplication, and then carefully group the like terms before combining them. Rushing through the problem is another big no-no. Take your time, work through each step methodically, and double-check your work as you go. It’s better to be slow and accurate than fast and wrong. And finally, don’t forget the basics of arithmetic. Simple addition, subtraction, multiplication, and division errors can sneak in and mess up your final answer. It might sound obvious, but it’s worth saying: double-check your calculations. By being aware of these common mistakes and taking steps to avoid them, you'll be well on your way to mastering polynomial multiplication. So, keep these tips in mind, and happy calculating!

Practice Problems

Okay, guys, time to put what we've learned into action! Practice makes perfect, especially when it comes to math. So, let's dive into some practice problems that are similar to our example, (3x - 6)(2x^2 - 7x + 1). Working through these will help you solidify your understanding and boost your confidence. Here's the first one: Try multiplying (2x + 3)(x^2 - 4x + 5). Take your time, remember the distributive property, and watch out for those tricky signs! Break it down step by step, just like we did earlier. Another great problem to tackle is (4x - 1)(3x^2 + 2x - 2). This one has similar terms to our original example, so it's excellent practice for honing your skills. Remember to combine like terms at the end to simplify your answer fully. If you want to mix things up a bit, how about trying (x + 5)(2x^2 - x - 3)? This one has slightly different coefficients and signs, which will challenge you to stay sharp. Don't forget to double-check each step to avoid those common mistakes we talked about. And here's one more for good measure: (3x - 2)(x^2 + 5x - 4). This problem will give you even more practice with the distributive property and combining like terms. As you work through these problems, think about each step carefully. Are you distributing correctly? Are you paying attention to the signs? Are you combining like terms accurately? The more you practice, the more natural these steps will become. And if you get stuck, don't worry! Go back and review the step-by-step solution we worked through earlier. Remember, math is a skill that improves with practice. So, grab a pencil, get comfortable, and let's conquer these practice problems together! You've got this!

Conclusion

Alright, guys, we've reached the end of our journey to understand the product of (3x - 6)(2x^2 - 7x + 1)! We've covered a lot of ground, from understanding the basics of polynomial multiplication to tackling the problem step-by-step and even looking at common mistakes to avoid. You've learned how to apply the distributive property like a pro, combine like terms with confidence, and steer clear of those pesky errors that can trip you up. Remember, multiplying polynomials is a fundamental skill in algebra, and mastering it opens doors to more complex mathematical concepts. It's not just about getting the right answer; it's about understanding the process and building a solid foundation for future math challenges. We started by breaking down the problem into manageable steps, making sure every term got its turn in the multiplication game. Then, we carefully combined like terms to simplify our expression. We also highlighted those common pitfalls, like forgetting to distribute or messing up the signs, so you can be extra vigilant in your own calculations. And, of course, we threw in some practice problems to help you solidify your skills and gain confidence. So, what's the takeaway here? Practice, patience, and attention to detail are your best friends when it comes to multiplying polynomials. Keep working at it, and don't be afraid to make mistakes – they're part of the learning process. Each time you tackle a problem, you're strengthening your understanding and building your math muscles. You've got the tools and the knowledge, so go out there and conquer those polynomials! And remember, if you ever need a refresher, this guide is here for you. Happy calculating, and keep rocking those math problems!