Simplified Sum Of Polynomials 3x²y²-2xy⁵ And -3x²y²+3x⁴y

Jul 16, 2025 by Jeany 57 views

Understanding Polynomial Sums The Simplified Sum of 3x²y²-2xy⁵ and -3x²y²+3x⁴y

In the realm of mathematics, polynomials form a fundamental concept. Polynomials are algebraic expressions comprising variables and coefficients, combined using addition, subtraction, and non-negative integer exponents. These expressions are the building blocks for more complex mathematical models and equations. This article delves into a specific problem involving the addition and simplification of polynomials. Specifically, we will explore the polynomials 3x²y² - 2xy⁵ and -3x²y² + 3x⁴y, meticulously simplifying their sum and categorizing the result based on its characteristics. Our primary goal is to determine the true nature of the resulting polynomial, identifying its type and degree, thereby enhancing our understanding of polynomial manipulation. By carefully examining the coefficients, variables, and exponents, we aim to provide a comprehensive analysis that sheds light on the intricacies of polynomial arithmetic.

Adding and Simplifying Polynomials

To begin, we embark on the journey of adding the polynomials 3x²y² - 2xy⁵ and -3x²y² + 3x⁴y. The cornerstone of this process lies in the identification and combination of like terms. Like terms are those that possess the same variables raised to the same powers. This principle ensures that we are only combining terms that are mathematically compatible, leading to an accurate simplification. Upon careful inspection of our polynomials, we notice that the terms 3x²y² and -3x²y² emerge as like terms. These terms share the same variables, x and y, each raised to the second power. The simplicity of this addition is striking, as 3x²y² and -3x²y² neatly cancel each other out. This cancellation is a crucial step in simplifying the expression, as it eliminates one set of terms and streamlines our polynomial. Following this cancellation, we are left with the remaining terms: -2xy⁵ and 3x⁴y. These terms, however, are not like terms. The first term, -2xy⁵, involves x raised to the first power and y raised to the fifth power, while the second term, 3x⁴y, presents x raised to the fourth power and y raised to the first power. Due to these differences in exponents, the terms cannot be combined any further. Thus, the simplified sum of our polynomials stands as -2xy⁵ + 3x⁴y, a concise and refined expression that forms the basis for our subsequent analysis.

Detailed Step-by-Step Addition

Write down the two polynomials: (3x²y² - 2xy⁵) + (-3x²y² + 3x⁴y)
Identify like terms: 3x²y² and -3x²y² are like terms.
Combine like terms: 3x²y² - 3x²y² = 0
Write down the remaining terms: -2xy⁵ + 3x⁴y
The simplified sum is: -2xy⁵ + 3x⁴y

Classifying the Simplified Sum

With the simplified sum -2xy⁵ + 3x⁴y in hand, our next endeavor is to classify this expression. Classifying polynomials involves identifying their type based on the number of terms they contain and determining their degree, a measure of their complexity. In our simplified sum, we observe two distinct terms: -2xy⁵ and 3x⁴y. A polynomial with exactly two terms is known as a binomial. This classification is derived from the prefix "bi-", signifying "two", and "nomial", referring to terms. Therefore, we can confidently categorize our simplified sum as a binomial. Now, we turn our attention to the degree of the polynomial. The degree of a polynomial is determined by the highest sum of the exponents of the variables within a single term. To ascertain this, we examine each term individually. In the term -2xy⁵, the variable x has an exponent of 1, and the variable y has an exponent of 5. Summing these exponents, we get 1 + 5 = 6. For the term 3x⁴y, the exponent of x is 4, and the exponent of y is 1, resulting in a sum of 4 + 1 = 5. Comparing the degrees of these two terms, we find that the highest degree is 6, which comes from the term -2xy⁵. Consequently, the degree of the entire polynomial -2xy⁵ + 3x⁴y is 6. This signifies that the simplified sum is a binomial of degree 6, providing a comprehensive classification of the polynomial's structure and complexity.

Determining the Degree of a Polynomial

Identify the exponents of the variables in each term.
Sum the exponents in each term.
The highest sum is the degree of the polynomial.

For -2xy⁵: The exponents are 1 (for x) and 5 (for y). The sum is 1 + 5 = 6. For 3x⁴y: The exponents are 4 (for x) and 1 (for y). The sum is 4 + 1 = 5. The highest sum is 6, so the degree of the polynomial is 6.

Analyzing the Options

Now that we have meticulously simplified the sum of the given polynomials and classified the result, we are well-equipped to analyze the provided options. Our simplified sum, -2xy⁵ + 3x⁴y, has been identified as a binomial with a degree of 6. This detailed analysis forms the basis for evaluating the accuracy of the given statements. Let's consider each option in turn:

Option A: The sum is a trinomial with a degree of 5. A trinomial is a polynomial with three terms. Our simplified sum has only two terms, making it a binomial, not a trinomial. Additionally, we determined that the degree of the polynomial is 6, not 5. Therefore, this option is incorrect.
Option B: The sum is a trinomial with a degree of 6. While this option correctly identifies the degree as 6, it incorrectly classifies the polynomial as a trinomial. As established earlier, our simplified sum is a binomial, consisting of two terms. Thus, this option is also incorrect.
Option C: The sum is a binomial with a degree of 6. This option accurately describes our simplified sum. We have definitively shown that the polynomial -2xy⁵ + 3x⁴y is a binomial, comprising two terms, and that its degree is 6, determined by the highest sum of the exponents in any term. Therefore, this option aligns perfectly with our analysis and stands as the correct answer.

By systematically working through the simplification and classification process, we have been able to confidently identify the true nature of the polynomial and select the correct option.

Detailed Analysis of Each Option

Option A: The sum is a trinomial with a degree of 5.
- Why it's incorrect: A trinomial has three terms. The simplified sum has two terms. The degree is 6, not 5.
Option B: The sum is a trinomial with a degree of 6.
- Why it's incorrect: Again, the term count is wrong. It's a binomial, not a trinomial, even though the degree is correctly identified as 6.
Option C: The sum is a binomial with a degree of 6.
- Why it's correct: The simplified sum has two terms, making it a binomial. The highest sum of exponents is 6, confirming the degree.

Conclusion

In conclusion, the journey of simplifying and classifying polynomials provides a valuable insight into the structure and characteristics of algebraic expressions. By systematically adding the polynomials 3x²y² - 2xy⁵ and -3x²y² + 3x⁴y, we arrived at the simplified sum -2xy⁵ + 3x⁴y. This expression, upon careful examination, was identified as a binomial, characterized by its two terms. Furthermore, the degree of the polynomial was determined to be 6, based on the highest sum of the exponents of the variables within a single term. Through this analysis, we were able to definitively conclude that option C, "The sum is a binomial with a degree of 6," is the correct answer. This exercise underscores the importance of meticulous simplification and classification in polynomial arithmetic, highlighting the fundamental principles that govern these mathematical constructs. The ability to accurately manipulate and categorize polynomials is crucial for success in more advanced mathematical studies, providing a solid foundation for tackling complex equations and models. Understanding the degree and type of a polynomial allows mathematicians and students alike to predict the behavior and properties of functions, making it an indispensable tool in various fields of science and engineering.

Key Takeaways

When adding polynomials, combine like terms by adding or subtracting their coefficients.
The degree of a polynomial is the highest sum of the exponents in any term.
Polynomials are classified by the number of terms (monomial, binomial, trinomial, etc.).
Accurate simplification and classification are essential for solving mathematical problems.

By mastering these fundamental concepts, one can confidently navigate the world of polynomials and algebraic expressions, paving the way for deeper mathematical understanding and proficiency.