Polynomial Sum 6s²t-2st² And 4s²t-3st² Binomial Degree Explained

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In the realm of mathematics, polynomials stand as fundamental expressions, weaving together variables and coefficients through the elegant operations of addition, subtraction, and multiplication. Understanding the behavior of polynomials, especially when they interact through addition, is crucial for mastering algebraic concepts. In this comprehensive exploration, we delve into the intricacies of polynomial addition, focusing on a specific example involving the polynomials 6s²t-2st² and 4s²t-3st². Our mission is to unravel the nature of their sum, meticulously examining its structure and characteristics to determine whether it emerges as a binomial or trinomial, and to precisely identify its degree.

Deciphering Polynomials: A Foundation for Understanding

Before embarking on the journey of adding polynomials, it is essential to establish a firm understanding of their fundamental components. A polynomial, in its essence, is an expression constructed from variables, constants, and exponents, interwoven through the operations of addition, subtraction, and multiplication. The building blocks of a polynomial are terms, each consisting of a coefficient (a numerical factor) and one or more variables raised to non-negative integer powers. For instance, in the polynomial 6s²t-2st², the terms are 6s²t and -2st².

The degree of a term is determined by summing the exponents of the variables within that term. In the term 6s²t, the variable 's' has an exponent of 2, and the variable 't' has an exponent of 1 (implicitly). Therefore, the degree of this term is 2 + 1 = 3. Similarly, the degree of the term -2st² is 1 + 2 = 3. The degree of the polynomial itself is defined as the highest degree among all its terms. In our example, both terms have a degree of 3, making the degree of the polynomial 6s²t-2st² equal to 3.

Polynomials are further classified based on the number of terms they contain. A monomial consists of a single term, a binomial comprises two terms, and a trinomial is composed of three terms. Polynomials with more than three terms are simply referred to as polynomials.

Adding Polynomials: A Step-by-Step Approach

Adding polynomials is a straightforward process that hinges on the concept of like terms. Like terms are those that share the same variables raised to the same powers. For example, 3x²y and -5x²y are like terms because they both have the variables 'x' raised to the power of 2 and 'y' raised to the power of 1. However, 3x²y and 3xy² are not like terms because the exponents of the variables are different.

The procedure for adding polynomials involves the following steps:

  1. Identify like terms: Carefully examine the polynomials and group together the terms that have the same variables raised to the same powers.
  2. Combine like terms: For each group of like terms, add their coefficients while keeping the variables and exponents unchanged. Remember that adding like terms is essentially combining their numerical coefficients.

Now, let's apply these steps to our specific example of adding the polynomials 6s²t-2st² and 4s²t-3st².

Unveiling the Sum: Adding 6s²t-2st² and 4s²t-3st²

To add the polynomials 6s²t-2st² and 4s²t-3st², we meticulously follow the steps outlined above:

  1. Identify like terms:

    • The terms 6s²t and 4s²t are like terms because they both contain the variables 's' raised to the power of 2 and 't' raised to the power of 1.
    • The terms -2st² and -3st² are like terms because they both contain the variables 's' raised to the power of 1 and 't' raised to the power of 2.

  2. Combine like terms:

    • Adding the like terms 6s²t and 4s²t yields (6 + 4)s²t = 10s²t.
    • Adding the like terms -2st² and -3st² yields (-2 - 3)st² = -5st².

Therefore, the sum of the polynomials 6s²t-2st² and 4s²t-3st² is 10s²t - 5st².

Analyzing the Sum: Binomial or Trinomial?

Now that we have obtained the sum, 10s²t - 5st², we can analyze its structure to determine whether it is a binomial or a trinomial. A binomial, by definition, consists of two terms, while a trinomial comprises three terms. In our case, the sum 10s²t - 5st² clearly has two terms: 10s²t and -5st². Consequently, we can confidently conclude that the sum is a binomial.

Determining the Degree: Unveiling the Highest Power

To ascertain the degree of the sum, 10s²t - 5st², we need to identify the highest degree among its terms. The degree of a term is calculated by summing the exponents of its variables.

  • The term 10s²t has a degree of 2 + 1 = 3 (the exponent of 's' is 2, and the exponent of 't' is 1).
  • The term -5st² also has a degree of 1 + 2 = 3 (the exponent of 's' is 1, and the exponent of 't' is 2).

Since both terms have a degree of 3, the degree of the polynomial sum 10s²t - 5st² is 3. Therefore, the sum is a binomial with a degree of 3.

Conclusion: The Sum is a Binomial with a Degree of 3

In this comprehensive exploration, we embarked on a journey to decipher the nature of the sum of the polynomials 6s²t-2st² and 4s²t-3st². Through a meticulous step-by-step approach, we successfully added the polynomials, identified like terms, and combined them to arrive at the sum 10s²t - 5st². Our analysis revealed that this sum is a binomial, characterized by its two terms. Furthermore, by examining the degrees of the individual terms, we determined that the degree of the sum is 3. Therefore, the definitive answer to the question of what is true about the sum of the two polynomials is that the sum is a binomial with a degree of 3. This detailed exploration not only provides a clear understanding of the specific example but also reinforces the fundamental principles of polynomial addition and analysis.

By understanding the fundamental principles of polynomial operations, we pave the way for tackling more complex algebraic challenges. The ability to confidently manipulate and analyze polynomials is an invaluable asset in various fields, including engineering, physics, and computer science. As we continue our mathematical journey, let us embrace the elegance and power of polynomials, recognizing them as essential tools for unlocking the secrets of the mathematical universe.