Polynomial Sum & Classification: A Step-by-Step Guide

Hey everyone, let's dive into the world of polynomials! Today, we're going to tackle a problem that involves both finding the sum of polynomials and classifying them based on their degree and the number of terms. It's like a fun puzzle, and I'll walk you through it step by step. We'll be looking at the expression: $3 n^2\left(5 n^2-2 n+1\right)+\left(4 n^2-11 n^4-9\right)$. So, grab your pencils and let's get started!

Understanding the Basics: What are Polynomials?

Before we jump into the problem, let's quickly recap what a polynomial is, because understanding the basics is super important. Think of a polynomial as a mathematical expression made up of terms. These terms can be constants (like the number 5), variables (like n), or variables raised to non-negative integer powers (like n² or n³), all combined using addition, subtraction, and multiplication.

Each part of a polynomial, separated by plus or minus signs, is called a term. For instance, in the polynomial $3n^2 - 2n + 1$, the terms are $3n^2$, $-2n$, and $1$. The degree of a term is the exponent of the variable in that term. If a term doesn't have a variable, its degree is 0. The degree of the polynomial is the highest degree of any of its terms. So, in $3n^2 - 2n + 1$, the degree of the polynomial is 2 (because of the $3n^2$ term). The number of terms is simply the count of the individual terms in the polynomial. Let's not forget the importance of combining like terms when simplifying a polynomial expression. Like terms are those that have the same variable raised to the same power. When simplifying, we combine the coefficients of like terms. For example, in the expression $2x^2 + 3x^2$, the like terms are $2x^2$ and $3x^2$. Combining them, we get $5x^2$. So, remember, these fundamental concepts are your building blocks for solving more complex polynomial problems. Now that we've refreshed our memories, let's tackle the given polynomial expression step by step, making sure we apply these concepts to find the sum and classify it effectively. Trust me, it's going to be a fun ride, and you'll see how these concepts come alive when we apply them to our specific example! Are you ready to dive in?

Step-by-Step Solution: Finding the Sum of the Polynomial

Alright, buckle up, because now we are going to calculate the sum of the polynomial. First, we need to simplify the given expression: $3 n^2\left(5 n^2-2 n+1\right)+\left(4 n^2-11 n^4-9\right)$. We will start by distributing the $3n^2$ across the terms inside the parentheses in the first part of the expression. This means multiplying $3n^2$ by each term within the parentheses. So, we'll multiply $3n^2$ by $5n^2$, by $-2n$, and by $1$. Let's break it down: $3n^2 * 5n^2 = 15n^4$, $3n^2 * -2n = -6n^3$, and $3n^2 * 1 = 3n^2$. Now, our expression looks like this: $15n^4 - 6n^3 + 3n^2 + (4n^2 - 11n^4 - 9)$. Next, let's remove the parentheses and combine like terms. This involves grouping the terms that have the same variable raised to the same power. We can see that we have two $n^4$ terms ($15n^4$ and $-11n^4$) and two $n^2$ terms ($3n^2$ and $4n^2$). Let's combine those like terms. For the $n^4$ terms, we have $15n^4 - 11n^4 = 4n^4$. For the $n^2$ terms, we have $3n^2 + 4n^2 = 7n^2$. Now, rewrite the expression with the combined like terms. The simplified expression is $4n^4 - 6n^3 + 7n^2 - 9$. So, we have successfully found the sum of the given polynomials! Now that we have the simplified form, let's move on to the fun part: classifying it based on its degree and number of terms. This is like the cherry on top, folks! We're almost there, keep going!

Classifying the Polynomial: Degree and Number of Terms

Alright, we've done the heavy lifting by finding the sum. Now, let's classify the polynomial we got, $4n^4 - 6n^3 + 7n^2 - 9$, based on its degree and the number of terms. Remember, the degree of a polynomial is the highest power of the variable in the expression. Looking at our simplified expression, the highest power of n is 4 (in the term $4n^4$). Therefore, the degree of this polynomial is 4. This means our polynomial is a 4th degree polynomial. Pretty straightforward, right? Now, let's look at the number of terms. The terms are the individual parts of the polynomial separated by plus or minus signs. In our expression, we have $4n^4$, $-6n^3$, $7n^2$, and $-9$. That's four distinct terms. Therefore, this polynomial has 4 terms. So, we've successfully classified the polynomial. It's a 4th degree polynomial with 4 terms.

To recap, the degree of the polynomial tells us the highest power of the variable, and the number of terms tells us how many parts make up the polynomial. Easy peasy! In summary, we have found the sum of the polynomials and classified it as a 4th degree polynomial with 4 terms. You've got it, guys! We're making progress. Now, let's compare our answer to the multiple-choice options, just to be sure.

Comparing with the Answer Choices

Okay, let's go back and check the options in the original question. Remember, the question presented us with a simplified polynomial: $4n^4 - 6n^3 + 7n^2 - 9$. And we know now that the polynomial is a 4th-degree polynomial with 4 terms. Now we must compare this with the given choices, our answer is definitely not A, because the degree of the polynomial is 4, not 3. Therefore, the correct answer is definitely not A. Since we've determined that the polynomial is a 4th degree polynomial with 4 terms. None of the other options seem to fit our result. So, the correct answer is not present in the given options.

Conclusion: You Did It!

Awesome work, everyone! We've successfully found the sum of the given polynomials and classified the result based on degree and the number of terms. We broke down the problem step by step, from understanding the basics to simplifying the expression, and finally classifying it. The key takeaways are to always remember to distribute, combine like terms, and then identify the highest power (degree) and count the individual terms. Remember, practice makes perfect! The more you work with polynomials, the easier it will become. If you're a bit confused, don't worry, just go back and review the steps. Keep practicing, and you'll be a polynomial pro in no time! Now, go out there and conquer those polynomials! Thanks for joining me today. Keep up the great work! If you have any questions, feel free to ask. Cheers!