Polynomial Sum: Degree And Type Explained

Nov 14, 2025 by Alex Johnson 42 views

Let's dive into understanding the sum of two polynomials: $6s^2t - 2st^2$ and $4s^2t - 3st^2$ . We'll break down the process step by step to determine the correct description of the resulting polynomial.

Understanding Polynomial Basics

Before we jump into adding these polynomials, let's clarify some key terms. A polynomial is an expression consisting of variables (like s and t) and coefficients, combined using addition, subtraction, and non-negative integer exponents. The degree of a term in a polynomial is the sum of the exponents of the variables in that term. The degree of the polynomial itself is the highest degree of any of its terms.

A binomial is a polynomial with two terms.
A trinomial is a polynomial with three terms.

Now that we have the basics down, let's tackle the problem.

Adding the Polynomials

To find the sum of the given polynomials, we combine like terms. Like terms are terms that have the same variables raised to the same powers. In this case, we have:

$6s^2t - 2st^2$ and $4s^2t - 3st^2$

We can add these polynomials by combining the terms with $s^2t$ and the terms with $st^2$ :

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

This simplifies to:

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

Analyzing the Result

Now that we have the sum, $10s^2t - 5st^2$ , let's analyze its properties.

Number of Terms

The resulting polynomial has two terms: $10s^2t$ and $-5st^2$ . Therefore, it is a binomial.

Determining the Degree

To find the degree of the polynomial, we need to find the degree of each term and then take the highest degree.

The degree of the term $10s^2t$ is $2 + 1 = 3$ (since the exponent of s is 2 and the exponent of t is 1).
The degree of the term $-5st^2$ is $1 + 2 = 3$ (since 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 is 3.

Conclusion

Based on our analysis, the sum of the polynomials $6s^2t - 2st^2$ and $4s^2t - 3st^2$ is a binomial with a degree of 3. Therefore, the correct answer is:

B. The sum is a binomial with a degree of 3.

Understanding how to add polynomials and determine their degree and type is fundamental in algebra. This exercise not only reinforces these concepts but also enhances problem-solving skills in mathematical contexts.

Further Exploration of Polynomials

To deepen your understanding, consider exploring additional resources and practicing more problems. Here are some areas to focus on:

Different Types of Polynomials

Familiarize yourself with different types of polynomials based on the number of terms (monomial, binomial, trinomial) and their degrees (linear, quadratic, cubic, etc.). Understanding these classifications can help you quickly identify and work with various polynomial expressions.

Operations with Polynomials

Practice adding, subtracting, multiplying, and dividing polynomials. Each operation has its own set of rules and techniques. For example, multiplying polynomials often involves using the distributive property and combining like terms.

Factoring Polynomials

Factoring is the process of breaking down a polynomial into simpler expressions (factors). This is a crucial skill in algebra and is used to solve equations, simplify expressions, and analyze functions.

Polynomial Equations

Learn how to solve polynomial equations, which are equations involving polynomials. Techniques for solving these equations include factoring, using the quadratic formula, and applying numerical methods.

Real-World Applications

Explore real-world applications of polynomials in fields such as physics, engineering, economics, and computer science. Understanding these applications can help you appreciate the practical significance of polynomials.

Advanced Topics

If you're interested in more advanced topics, you can explore concepts such as polynomial functions, polynomial graphs, polynomial inequalities, and polynomial regression.

Tips for Success

Here are some tips to help you succeed in working with polynomials:

Practice Regularly: The more you practice, the better you'll become at recognizing patterns and applying the correct techniques.
Review the Basics: Make sure you have a strong understanding of the fundamental concepts, such as variables, exponents, coefficients, and operations.
Break Down Problems: Break down complex problems into smaller, more manageable steps. This can help you avoid mistakes and stay organized.
Check Your Work: Always check your work to make sure you haven't made any errors. You can also use a calculator or computer algebra system to verify your results.
Seek Help When Needed: Don't be afraid to ask for help from teachers, tutors, or online resources if you're struggling with a concept.

By following these tips and continuously practicing, you can improve your skills and confidence in working with polynomials.