Multiplying Polynomials: A Step-by-Step Guide

Hey guys! Let's dive into the world of algebra and tackle a fun problem: multiplying polynomials. In this article, we'll break down the process step-by-step, ensuring you understand how to find the product of expressions like $\left(y^2+3 y+7\right)\left(8 y^2+y+1\right)$. We'll also explore the different answer choices and pinpoint the correct one. So, grab your pencils, and let's get started! Understanding this concept is key to solving more complex algebraic problems, so pay close attention. We will be using the distributive property, a fundamental concept in algebra, to solve this problem, so let's get into it.

The Problem: Unveiling the Product

Our mission, should we choose to accept it, is to find the product of the following expression: $\left(y^2+3 y+7\right)\left(8 y^2+y+1\right)$. What does this even mean? Simply put, we need to multiply these two polynomials together. Polynomials are expressions with variables and coefficients, combined using addition, subtraction, and multiplication. Don't worry, it's not as scary as it sounds! It's actually a pretty straightforward process once you get the hang of it. The key to solving this problem lies in understanding the distributive property. This property states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. We'll use this principle to expand our expression. Now, let's look at the answer choices provided to find out the correct result. Remember, attention to detail is crucial here, as a small mistake can lead to an incorrect answer. Are you ready? Let's begin the exciting journey of finding the correct answer.

Step-by-Step Solution: Multiplying with Precision

Alright, let's get down to business and figure out how to multiply these polynomials. Here's how we'll do it, step by step: We'll start by multiplying each term in the first polynomial (y² + 3y + 7) by each term in the second polynomial (8y² + y + 1). Let's break it down:

1. Multiply y² by each term in (8y² + y + 1):

   • y² * 8y² = 8y⁴ (Remember, when multiplying exponents, you add them: y² * y² = y^(2+2) = y⁴)
   • y² * y = y³ (y² * y¹ = y^(2+1) = y³)
   • y² * 1 = y²

2. Multiply 3y by each term in (8y² + y + 1):

   • 3y * 8y² = 24y³
   • 3y * y = 3y²
   • 3y * 1 = 3y

3. Multiply 7 by each term in (8y² + y + 1):

   • 7 * 8y² = 56y²
   • 7 * y = 7y
   • 7 * 1 = 7

Now, let's put it all together and sum up the results: 8y⁴ + y³ + y² + 24y³ + 3y² + 3y + 56y² + 7y + 7. The next step is to combine like terms. This means adding or subtracting terms that have the same variable and exponent. By carefully following these steps, you will be able to multiply polynomials efficiently. It's all about keeping track of each term and performing the calculations accurately. Let's make sure we collect all the similar terms and add them together.

Combining Like Terms: Simplifying the Expression

Now that we've done all the multiplications, it's time to simplify our expression by combining like terms. This is where we gather all the terms with the same power of y and add their coefficients. Let's do it: We have $8y^4$, which doesn't have any like terms. So, we'll keep that as is. Then, we have $y^3$ and $24y^3$. Combining them, we get $y^3 + 24y^3 = 25y^3$. Next, we've got $y^2$, $3y^2$, and $56y^2$. Combining those gives us $y^2 + 3y^2 + 56y^2 = 60y^2$. After that, we have $3y$ and $7y$, which combine to $3y + 7y = 10y$. Lastly, we have the constant term, $7$, which remains as it is. So, after combining like terms, our expression simplifies to: $8y^4 + 25y^3 + 60y^2 + 10y + 7$. Keep in mind the order of operations and the exponent rules for each variable when multiplying and combining like terms. You are well on your way to mastering polynomial multiplication! With this step, we've gone from a complex expression to a simplified form, making it much easier to identify the correct answer choice. We are almost at the final step.

Identifying the Correct Answer: The Final Verdict

Alright, guys, we've done all the hard work! Now, we just need to match our simplified expression to one of the answer choices. Remember, our simplified expression is: $8y^4 + 25y^3 + 60y^2 + 10y + 7$. Now, let's look at the multiple-choice options:

• A. $8y^4 + 24y^3 + 60y^2 + 10y + 7$: This is incorrect because the coefficient of y³ is different.
• B. $8y^4 + 25y^3 + 4y^2 + 10y + 7$: Incorrect, as the coefficient of y² is incorrect.
• C. $8y^4 + 25y^3 + 60y^2 + 7y + 7$: Incorrect, because the coefficient of y is different.
• D. $8y^4 + 25y^3 + 60y + 10y + 7$: Incorrect, because the exponents are not correct.

As we can see, by careful observation and calculation, answer choice C. $8y^4 + 25y^3 + 60y^2 + 10y + 7$ is the correct answer. We've successfully multiplied the polynomials and identified the correct product. Congrats on reaching the end of the problem. You did a great job!

Conclusion: Mastering Polynomial Multiplication

And there you have it! We've successfully multiplied two polynomials and found the correct answer. We started with the expression $\left(y^2+3 y+7\right)\left(8 y^2+y+1\right)$, and through careful application of the distributive property and combining like terms, we arrived at the solution. The key takeaways from this problem are: Always remember the distributive property. Take your time, and be careful with your calculations. Double-check your work to avoid any silly mistakes. With practice, you'll become a pro at multiplying polynomials! Keep practicing, and you'll be acing these problems in no time. This skill is crucial for future algebraic concepts. Keep up the amazing work.

I hope this step-by-step guide has been helpful. Keep practicing, and you'll become a pro at multiplying polynomials. Remember to always double-check your calculations, and don't be afraid to break down the problem into smaller, more manageable steps. You've got this, and with practice, you'll be conquering polynomial multiplication like a boss! Feel free to revisit this guide whenever you need a refresher. Thanks for joining me on this math adventure, and happy calculating!