Simplify Polynomials: A Step-by-Step Guide

Hey guys! Let's dive into simplifying this polynomial expression. We've got $(3x - 4)(2x^2 - 3x + 5)$, and our mission is to multiply these two expressions together and combine like terms. This kind of problem is super common in algebra, so mastering it will definitely give you a leg up. We'll go through each step meticulously, so you can follow along and understand exactly how it's done. Think of it like baking a cake – each ingredient (or term) needs to be added in the right order and mixed properly to get the perfect result. So, grab your mathematical whisks, and let’s get started!

Understanding Polynomial Multiplication

Before we jump into the specifics of this problem, let's quickly recap what polynomial multiplication is all about. Polynomials are expressions made up of variables and coefficients, combined using addition, subtraction, and multiplication. When we multiply two polynomials, we're essentially using the distributive property – which, if you remember, is like making sure every term in the first polynomial gets multiplied by every term in the second polynomial. It’s kind of like hosting a party where each guest (term) needs to shake hands with every other guest. So, if you have two polynomials, say (A + B) and (C + D), you'd multiply A by both C and D, and then multiply B by both C and D. This gives you AC + AD + BC + BD. Simple, right? The key is to keep track of your terms and signs, and you'll be golden!

When dealing with larger polynomials, like the ones in our problem, it’s crucial to stay organized. A methodical approach ensures that no term is left behind and that like terms are correctly combined. Imagine trying to sort a massive pile of laundry – you wouldn't just throw everything together, would you? You'd sort by color, type, and so on. Similarly, in polynomial multiplication, we multiply systematically and then group like terms together. This not only makes the process less error-prone but also helps in understanding the structure of the resulting polynomial. So, let's put on our organizational hats and get ready to multiply!

Now, let’s apply this knowledge to our specific problem. We have $(3x - 4)(2x^2 - 3x + 5)$. This means we need to multiply each term in the first binomial $(3x - 4)$ by each term in the trinomial $(2x^2 - 3x + 5)$. It might seem a bit daunting at first, but breaking it down step by step makes it much more manageable. We'll start by multiplying $3x$ by each term in the trinomial, and then we'll do the same with $-4$. Think of it as a mini-multiplication marathon where each term gets its moment in the spotlight. Ready to see how it's done? Let's move on to the next section where we'll start the actual multiplication!

Step-by-Step Multiplication

Okay, let's get down to the nitty-gritty and multiply these polynomials step by step. Remember, we're aiming to multiply $(3x - 4)$ by $(2x^2 - 3x + 5)$. We'll start by distributing $3x$ across the trinomial $(2x^2 - 3x + 5)$.

First, multiply $3x$ by $2x^2$. This gives us $(3x)(2x^2) = 6x^3$. Remember, when you multiply variables with exponents, you add the exponents. So, $x$ (which is $x^1$) times $x^2$ becomes $x^{1+2} = x^3$.

Next up, multiply $3x$ by $-3x$. This gives us $(3x)(-3x) = -9x^2$. Again, we add the exponents, so $x$ times $x$ is $x^2$, and the coefficients 3 and -3 multiply to -9.

Finally, multiply $3x$ by 5. This gives us $(3x)(5) = 15x$. Straightforward enough, right? We've now taken care of the first part of our distribution.

So, after distributing $3x$, we have $6x^3 - 9x^2 + 15x$. Now, we move on to the next term in our binomial, which is $-4$. We'll distribute $-4$ across the same trinomial $(2x^2 - 3x + 5)$.

Multiply $-4$ by $2x^2$. This gives us $(-4)(2x^2) = -8x^2$. Simple multiplication of the coefficients and keeping the $x^2$ term.

Next, multiply $-4$ by $-3x$. This gives us $(-4)(-3x) = 12x$. Remember, a negative times a negative is a positive, so -4 times -3 is 12.

Lastly, multiply $-4$ by 5. This gives us $(-4)(5) = -20$. A straightforward multiplication here.

So, after distributing $-4$, we have $-8x^2 + 12x - 20$. Now we have all the pieces of our puzzle. We multiplied $3x$ by the trinomial and $-4$ by the trinomial. The next step? You guessed it – combining like terms!

Combining Like Terms

Alright, we've done the hard work of multiplying everything out. Now comes the satisfying part: combining like terms. This is where we gather all the terms that have the same variable and exponent and add or subtract their coefficients. It’s like sorting your socks – you put all the pairs together, right? We're doing the same thing here, just with algebraic terms.

So, let’s look at what we have after our distribution: $6x^3 - 9x^2 + 15x - 8x^2 + 12x - 20$. Now, let’s identify the like terms.

First, we have the $x^3$ terms. In our expression, we only have one $x^3$ term, which is $6x^3$. So, that one is all by itself for now.

Next, let’s look at the $x^2$ terms. We have $-9x^2$ and $-8x^2$. These are like terms because they both have $x^2$. To combine them, we add their coefficients: $-9 + (-8) = -17$. So, $-9x^2 - 8x^2$ becomes $-17x^2$.

Now, let’s move on to the $x$ terms. We have $15x$ and $12x$. These are also like terms. We add their coefficients: $15 + 12 = 27$. So, $15x + 12x$ becomes $27x$.

Finally, we have the constant terms, which are just numbers without any variables. In our expression, we only have one constant term, which is $-20$. So, it remains as $-20$.

Now, let’s put it all together. We have $6x^3$, $-17x^2$, $27x$, and $-20$. Combining these gives us $6x^3 - 17x^2 + 27x - 20$. And there you have it! We've simplified the expression by combining like terms.

So, after all that multiplying and combining, we’ve arrived at our simplified polynomial. Let’s take a moment to appreciate the journey. We started with a product of a binomial and a trinomial, distributed each term carefully, and then grouped like terms to simplify the expression. It’s like building a house – each step is crucial to the final result. And just like a well-built house, our simplified polynomial is now in its most organized and understandable form.

The Final Answer

Okay, guys, we've reached the final step! After all the multiplying and combining like terms, we've simplified the expression $(3x - 4)(2x^2 - 3x + 5)$. Let's recap what we did:

1. We distributed $3x$ across $(2x^2 - 3x + 5)$, which gave us $6x^3 - 9x^2 + 15x$.
2. Then, we distributed $-4$ across $(2x^2 - 3x + 5)$, which gave us $-8x^2 + 12x - 20$.
3. Next, we combined all the terms: $6x^3 - 9x^2 + 15x - 8x^2 + 12x - 20$.
4. Finally, we grouped like terms and added their coefficients: $6x^3 + (-9x^2 - 8x^2) + (15x + 12x) - 20$.

This simplified to $6x^3 - 17x^2 + 27x - 20$.

So, the final answer is $6x^3 - 17x^2 + 27x - 20$.

Looking back at the options, we can see that this matches option A. So, A is the correct answer!

Congratulations, we've successfully simplified the given expression! This type of problem is a fundamental skill in algebra, and you've now got another tool in your mathematical toolkit. Remember, the key is to take it step by step, stay organized, and double-check your work. You’ve got this! Polynomial multiplication might seem daunting at first, but with practice, it becomes second nature. Think of each problem as a puzzle – a fun challenge that you can solve with the right techniques. And who knows, maybe you’ll even start enjoying these algebraic adventures!

Practice Makes Perfect

Now that we've walked through this problem together, the best way to really master polynomial multiplication is to practice, practice, practice! The more you work through these types of problems, the more comfortable and confident you'll become. It’s like learning to ride a bike – you might wobble a bit at first, but eventually, you’ll be cruising along smoothly. So, let’s talk about how you can get in some solid practice.

First off, try revisiting this problem on your own. Work through the steps without looking at our explanation and see if you can arrive at the same answer. This is a great way to reinforce what you've learned and identify any areas where you might need a little extra help. It’s like taking a practice lap around the track before the big race – it helps you get a feel for the course.

Next, seek out similar problems. Textbooks, online resources, and worksheets are fantastic sources of practice questions. Look for problems that involve multiplying binomials and trinomials, and gradually increase the complexity as you get more comfortable. Think of it as leveling up in a game – you start with the easy levels and gradually work your way up to the more challenging ones.

Another great strategy is to break down the process into smaller steps. If you’re struggling with the entire multiplication process, focus on one aspect at a time. For example, you could practice just distributing terms, or just combining like terms. By mastering each individual step, the overall process becomes much more manageable. It’s like learning a dance routine – you practice each move separately before putting it all together.

Don't be afraid to make mistakes! Mistakes are a natural part of the learning process. When you make a mistake, take the time to understand why you made it and how you can avoid it in the future. It’s like troubleshooting a recipe – if something doesn't turn out right, you figure out what went wrong and adjust your approach next time. Keep a positive attitude and remember that every mistake is an opportunity to learn and grow.

Finally, consider working with a study group or a tutor. Explaining the concepts to someone else can solidify your own understanding, and getting feedback from others can help you identify areas for improvement. It’s like having a workout buddy – you can motivate each other and stay on track. So, grab a friend, hit the books, and keep those polynomials coming! With consistent practice, you'll be simplifying expressions like a pro in no time. Keep up the great work, guys!