Understanding The Sum Of Polynomials A Detailed Explanation

In mathematics, polynomials are fundamental algebraic expressions that play a crucial role in various fields, from basic algebra to advanced calculus and beyond. Understanding the properties and operations involving polynomials is essential for students and professionals alike. One common operation is the addition of polynomials, which combines like terms to produce a simplified expression. In this article, we will delve into the process of adding two given polynomials and analyze the resulting sum to determine its characteristics. Specifically, we will examine the polynomials $6s^2t - 2st^2$ and $4s^2t - 3st^2$, add them together, and then identify the correct description of the resulting polynomial from the provided options. This exploration will not only reinforce the mechanics of polynomial addition but also enhance our understanding of polynomial classification based on the number of terms and degree.

Understanding Polynomials

Before we dive into the specific problem, let's take a moment to review the basics of polynomials. A polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. Polynomials can be classified based on the number of terms they contain. A monomial has one term, a binomial has two terms, a trinomial has three terms, and so on. For instance, $5x^2$ is a monomial, $3x + 2$ is a binomial, and $x^2 - 4x + 7$ is a trinomial. Additionally, polynomials are characterized by their degree, which is the highest power of the variable in the polynomial. In a multivariable polynomial, the degree of a term is the sum of the exponents of the variables in that term, and the degree of the polynomial is the highest degree among all its terms.

For example, consider the polynomial $7x^3 - 2x^2 + 5x - 1$. This is a polynomial in one variable, $x$. The terms are $7x^3$, $-2x^2$, $5x$, and $-1$. The coefficients are 7, -2, 5, and -1. The degrees of the terms are 3, 2, 1, and 0, respectively. Therefore, the degree of the polynomial is 3, which is the highest degree among its terms. Now, consider the polynomial $6s^2t - 2st^2$. This is a polynomial in two variables, $s$ and $t$. The degree of the term $6s^2t$ is $2 + 1 = 3$, and the degree of the term $-2st^2$ is $1 + 2 = 3$. Thus, the degree of the polynomial is 3. Understanding these basic concepts is crucial for performing operations on polynomials and classifying them correctly.

Adding the Polynomials

Now, let's focus on the given polynomials: $6s^2t - 2st^2$ and $4s^2t - 3st^2$. To add these polynomials, we need to combine like terms. Like terms are terms that have the same variables raised to the same powers. In this case, $6s^2t$ and $4s^2t$ are like terms, as both have $s$ raised to the power of 2 and $t$ raised to the power of 1. Similarly, $-2st^2$ and $-3st^2$ are like terms, as both have $s$ raised to the power of 1 and $t$ raised to the power of 2. To add like terms, we simply add their coefficients while keeping the variables and their exponents the same. So, let's add the polynomials:

$(6s^2t - 2st^2) + (4s^2t - 3st^2)$

First, combine the $s^2t$ terms:

$6s^2t + 4s^2t = (6 + 4)s^2t = 10s^2t$

Next, combine the $st^2$ terms:

$-2st^2 - 3st^2 = (-2 - 3)st^2 = -5st^2$

Now, combine the results:

$10s^2t - 5st^2$

The sum of the two polynomials is $10s^2t - 5st^2$. This resulting polynomial has two terms: $10s^2t$ and $-5st^2$. Therefore, it is a binomial. The degree of the term $10s^2t$ is $2 + 1 = 3$, and the degree of the term $-5st^2$ is $1 + 2 = 3$. The degree of the polynomial is the highest degree among its terms, which is 3.

Analyzing the Sum

Now that we have the sum, $10s^2t - 5st^2$, let's analyze its characteristics and compare them to the given options. We have already determined that the sum is a binomial because it has two terms. We have also found that the degree of the sum is 3, as both terms have a degree of 3. Now, let's consider the given options:

A. The sum is a binomial with a degree of 2. B. The sum is a binomial with a degree of 3. C. The sum is a trinomial with a degree of 3.

Comparing our findings with the options, we can see that option B, "The sum is a binomial with a degree of 3," accurately describes the sum we obtained. The sum $10s^2t - 5st^2$ indeed has two terms, making it a binomial, and its degree is 3, as we calculated. Option A is incorrect because the degree is 3, not 2. Option C is incorrect because the sum is a binomial, not a trinomial. Therefore, the correct answer is B.

Conclusion

In this article, we explored the process of adding two polynomials and analyzing the resulting sum. We added the polynomials $6s^2t - 2st^2$ and $4s^2t - 3st^2$ by combining like terms, which resulted in the polynomial $10s^2t - 5st^2$. We then analyzed the sum and determined that it is a binomial with a degree of 3. This analysis led us to conclude that option B, "The sum is a binomial with a degree of 3," is the correct description of the sum. Understanding polynomial addition and the classification of polynomials based on the number of terms and degree is a fundamental skill in algebra. By practicing such problems, we reinforce our understanding of these concepts and improve our ability to manipulate algebraic expressions effectively. This exercise demonstrates the importance of accurately performing polynomial operations and correctly identifying the characteristics of the resulting expressions. Mastering these skills is crucial for success in more advanced mathematical topics and various real-world applications. The ability to add polynomials, determine their degree, and classify them is not only essential for academic pursuits but also for practical problem-solving in fields such as engineering, computer science, and economics. Therefore, continuous practice and a solid understanding of these concepts are key to mathematical proficiency.