Degree Of Sum And Difference Of Polynomials Explained

When dealing with polynomials, one fundamental concept is the degree of a polynomial, which significantly influences its behavior and properties. To accurately analyze and manipulate polynomials, it's essential to understand how the degree changes when performing operations such as addition and subtraction. This article delves into determining the degree of the sum and difference of two given polynomials: $3x^5y - 2x^3y^4 - 7xy^3$ and $-8x^5y + 2x^3y^4 + xy^3$. We will explore the underlying principles of polynomial degrees and apply them step-by-step to arrive at the correct answer. Understanding the degree helps to classify polynomials, predict their end behavior in graphs, and simplify algebraic manipulations. The degree of the polynomial is defined as the highest sum of the exponents of the variables in any term of the polynomial. A term is a product of constants and variables, and in polynomials, these variables have non-negative integer exponents. Polynomials play a critical role in various mathematical and scientific applications, ranging from curve fitting in data analysis to modeling physical systems in engineering. Therefore, mastering polynomial operations is crucial for anyone looking to advance in these fields. Our focus here is on determining the degrees resulting from adding and subtracting polynomials, which lays the groundwork for more complex polynomial operations such as multiplication and division. This article will not only provide the solution to the specific problem but also offer insights into the general rules governing polynomial degrees under addition and subtraction. This approach enhances your overall understanding and ability to tackle similar problems with confidence. We aim to clarify the confusion often encountered while dealing with these algebraic concepts, making polynomial manipulation a more straightforward task. Let's break down the process step by step, beginning with a brief review of the definition of polynomial degree and how to find it for individual terms and entire polynomials.

Understanding Polynomial Degrees

In polynomial algebra, the degree of a term is the sum of the exponents of its variables. For example, in the term $3x^5y$, the degree is $5 + 1 = 6$, because the exponent of $x$ is 5 and the exponent of $y$ is 1 (since $y$ is the same as $y^1$). To find the degree of an entire polynomial, we identify the term with the highest degree. Let’s consider the polynomial $3x^5y - 2x^3y^4 - 7xy^3$. We examine each term:

• The degree of $3x^5y$ is $5 + 1 = 6$.
• The degree of $-2x^3y^4$ is $3 + 4 = 7$.
• The degree of $-7xy^3$ is $1 + 3 = 4$.

The highest degree among these terms is 7, so the degree of the polynomial $3x^5y - 2x^3y^4 - 7xy^3$ is 7. Similarly, for the polynomial $-8x^5y + 2x^3y^4 + xy^3$, we calculate:

• The degree of $-8x^5y$ is $5 + 1 = 6$.
• The degree of $2x^3y^4$ is $3 + 4 = 7$.
• The degree of $xy^3$ is $1 + 3 = 4$.

Again, the highest degree is 7, making the degree of the polynomial $-8x^5y + 2x^3y^4 + xy^3$ also 7. These calculations form the foundation for understanding how the degrees interact when we perform addition and subtraction of polynomials. The key concept here is that when adding or subtracting polynomials, the resulting degree is primarily determined by the terms with the highest original degrees, unless these terms cancel each other out. This cancellation is a crucial aspect that we will explore in the following sections when we calculate the sum and difference of the given polynomials. Understanding this principle allows us to predict the degree of the resulting polynomial without needing to fully expand and simplify the expression. For example, if two polynomials of degree $n$ are added, the resulting polynomial will have a degree of at most $n$. However, if the leading terms (the terms with the highest degree) are additive inverses, the degree of the sum will be less than $n$. This leads us to the next step, where we explicitly perform the addition and subtraction to determine the actual degrees of the resulting polynomials. By understanding these fundamental concepts, we can effectively analyze and simplify polynomials, which are essential skills in algebra and beyond. Now, let's move on to calculating the sum and difference of the provided polynomials.

Calculating the Sum of the Polynomials

To find the sum of the polynomials $3x^5y - 2x^3y^4 - 7xy^3$ and $-8x^5y + 2x^3y^4 + xy^3$, we combine like terms. Like terms are terms that have the same variables raised to the same powers. This process involves adding the coefficients of like terms while keeping the variable parts unchanged. In this case, we align the like terms and perform the addition:

$(3x^5y - 2x^3y^4 - 7xy^3) + (-8x^5y + 2x^3y^4 + xy^3)$

First, group the like terms together:

$(3x^5y - 8x^5y) + (-2x^3y^4 + 2x^3y^4) + (-7xy^3 + xy^3)$

Now, add the coefficients of each group:

$(3 - 8)x^5y + (-2 + 2)x^3y^4 + (-7 + 1)xy^3$

This simplifies to:

$-5x^5y + 0x^3y^4 - 6xy^3$

Which further simplifies to:

$-5x^5y - 6xy^3$

Now, we determine the degree of the resulting polynomial, $-5x^5y - 6xy^3$. The degree of the term $-5x^5y$ is $5 + 1 = 6$, and the degree of the term $-6xy^3$ is $1 + 3 = 4$. The highest degree among these is 6. Therefore, the degree of the sum of the polynomials is 6. This outcome highlights an important principle in polynomial addition: the degree of the sum can be less than the degree of the individual polynomials if the leading terms cancel each other out. In our case, the $x^3y^4$ terms canceled each other, resulting in a lower degree for the sum. Understanding how such cancellations affect the degree is crucial for simplifying and analyzing polynomials. This step-by-step process not only provides the solution but also emphasizes the method of combining like terms, a fundamental skill in polynomial algebra. By mastering this technique, you can confidently tackle more complex problems involving polynomial addition and subtraction. Next, we will calculate the difference of the polynomials, applying a similar process but with subtraction instead of addition. This will further illustrate how polynomial operations affect the degree and structure of the resulting expressions.

Calculating the Difference of the Polynomials

To find the difference of the polynomials, we subtract the second polynomial from the first. This operation involves distributing the negative sign across the terms of the second polynomial and then combining like terms. The polynomials are $3x^5y - 2x^3y^4 - 7xy^3$ and $-8x^5y + 2x^3y^4 + xy^3$. So we need to calculate:

$(3x^5y - 2x^3y^4 - 7xy^3) - (-8x^5y + 2x^3y^4 + xy^3)$

First, distribute the negative sign to each term in the second polynomial:

$3x^5y - 2x^3y^4 - 7xy^3 + 8x^5y - 2x^3y^4 - xy^3$

Next, group the like terms together:

$(3x^5y + 8x^5y) + (-2x^3y^4 - 2x^3y^4) + (-7xy^3 - xy^3)$

Now, combine the coefficients of the like terms:

$(3 + 8)x^5y + (-2 - 2)x^3y^4 + (-7 - 1)xy^3$

This simplifies to:

$11x^5y - 4x^3y^4 - 8xy^3$

To determine the degree of the resulting polynomial $11x^5y - 4x^3y^4 - 8xy^3$, we find the degree of each term:

• The degree of $11x^5y$ is $5 + 1 = 6$.
• The degree of $-4x^3y^4$ is $3 + 4 = 7$.
• The degree of $-8xy^3$ is $1 + 3 = 4$.

The highest degree among these terms is 7. Therefore, the degree of the difference of the polynomials is 7. This result underscores that when subtracting polynomials, the degree of the resulting polynomial is determined by the highest degree terms that remain after the operation. Unlike addition, where terms might cancel out, subtraction often preserves the original highest degree unless a specific term becomes zero due to subtraction. By carefully applying the rules of polynomial subtraction and combining like terms, we can accurately determine the degree of the resulting polynomial. This process not only provides the answer but also enhances our understanding of how polynomial operations affect their structure and degree. Next, we will compare our findings for both the sum and the difference to determine which of the given options is correct. This comparative analysis will reinforce our understanding of the principles involved and help solidify the concepts of polynomial degrees.

Determining the Correct Option

After calculating both the sum and the difference of the given polynomials, we found that:

• The sum of the polynomials $3x^5y - 2x^3y^4 - 7xy^3$ and $-8x^5y + 2x^3y^4 + xy^3$ is $-5x^5y - 6xy^3$, which has a degree of 6.
• The difference of the polynomials is $11x^5y - 4x^3y^4 - 8xy^3$, which has a degree of 7.

Now, let's consider the options provided:

A. Both the sum and difference have a degree of 6. B. Both the sum and difference have a degree of 7. C. The sum has a degree of 6, and the difference has a degree of 7.

Based on our calculations, option C is the correct choice. The sum of the polynomials has a degree of 6, and the difference has a degree of 7. This final step of comparison and selection reinforces the importance of accurate calculations and a thorough understanding of polynomial operations. By systematically working through the problem—first understanding the basic concepts, then performing the addition and subtraction, and finally determining the degrees—we arrived at the correct answer. This approach is crucial for tackling similar problems in algebra and beyond. Understanding the impact of polynomial operations on their degrees helps in simplifying expressions, solving equations, and analyzing mathematical models. In summary, the process of finding the degree of the sum and difference of polynomials involves several key steps: identifying like terms, performing the addition or subtraction, simplifying the result, and then determining the highest degree term in the resulting polynomial. This comprehensive method ensures that you can confidently handle polynomial operations and accurately determine their degrees. Understanding the nuances of polynomial algebra, such as how terms cancel out during addition or how subtraction can alter degrees, is essential for success in more advanced mathematical topics. With a solid grasp of these concepts, you are well-equipped to tackle a wide range of problems involving polynomials.

Conclusion

In conclusion, when analyzing the sum and difference of the polynomials $3x^5y - 2x^3y^4 - 7xy^3$ and $-8x^5y + 2x^3y^4 + xy^3$, we determined that the sum has a degree of 6, while the difference has a degree of 7. This demonstrates the importance of carefully performing polynomial operations and understanding how these operations affect the degree of the resulting polynomials. The process involved combining like terms after either adding or subtracting the polynomials, and then identifying the term with the highest degree in the simplified expression. Our step-by-step approach highlighted key algebraic principles, including the distribution of negative signs during subtraction and the cancellation of terms during addition. By understanding these concepts, we can confidently manipulate polynomials and determine their degrees, which is crucial for solving algebraic problems and analyzing mathematical models. The degree of a polynomial is a fundamental property that influences its behavior and characteristics. Knowing how to calculate and interpret the degree helps in simplifying expressions, solving equations, and understanding the graphical representation of polynomial functions. Polynomials are essential in various fields, including mathematics, physics, engineering, and computer science, making their manipulation and analysis a valuable skill. By mastering the techniques discussed in this article, you can enhance your algebraic proficiency and apply these skills to a wide range of real-world applications. The ability to perform polynomial operations accurately and efficiently is a cornerstone of algebraic competence. As you continue your mathematical journey, the concepts and methods covered here will serve as a solid foundation for more advanced topics. This exploration of polynomial degrees not only answers the specific question but also provides a broader understanding of polynomial algebra, equipping you with the tools to tackle more complex problems with confidence. Therefore, understanding the nuances of polynomial addition, subtraction, and degree calculation is essential for mathematical proficiency and problem-solving in various scientific and technical domains.