Polynomial Differences Simplifying And Determining Degree

• A. The difference has 3 terms and a degree of 6.
• B. The difference has 4 terms and a degree of 6.
• C. The difference has 3 terms and a degree of 7.
• D. The difference has 4 terms and a degree of 7.

Deciphering polynomial expressions and their operations can often seem like navigating a complex maze, but with a systematic approach, we can unravel even the most intricate problems. In this article, we aim to dissect a specific problem concerning the difference of two polynomials and determine which statement accurately describes the result. Specifically, we are tasked with finding the simplified difference of the polynomials $6x^6 - x^3y^4 - 5xy^5$ and $4x^5y + 2x^3y^4 + 5xy^5$. This involves understanding key concepts such as terms, degrees, and the process of simplification in polynomial arithmetic. To accurately identify the correct answer, we will meticulously walk through the steps of subtracting the given polynomials, combining like terms, and then analyzing the resulting expression. Our goal is to not only solve the problem at hand but also to elucidate the underlying principles of polynomial manipulation, thereby providing a comprehensive understanding for anyone tackling similar mathematical challenges. So, let's embark on this mathematical journey and demystify the world of polynomials.

Subtracting the Polynomials

To begin, we need to subtract the second polynomial from the first. This involves carefully distributing the negative sign across the terms of the second polynomial and then combining like terms. This is a critical step where attention to detail is paramount, as a simple sign error can lead to an incorrect answer. Before we dive into the subtraction, let's clearly state the two polynomials we are working with:

Polynomial 1: $6x^6 - x^3y^4 - 5xy^5$ Polynomial 2: $4x^5y + 2x^3y^4 + 5xy^5$

The operation we need to perform is (Polynomial 1) - (Polynomial 2), which translates to:

$(6x^6 - x^3y^4 - 5xy^5) - (4x^5y + 2x^3y^4 + 5xy^5)$

Now, let's distribute the negative sign across the terms of the second polynomial:

$6x^6 - x^3y^4 - 5xy^5 - 4x^5y - 2x^3y^4 - 5xy^5$

With the negative sign properly distributed, our next task is to identify and combine like terms. Like terms are terms that have the same variables raised to the same powers. In our expression, we can see terms involving $x^3y^4$ and $xy^5$. The term $6x^6$ and $-4x^5y$ don't have like terms and will remain as they are in the simplified expression. The process of combining like terms is a fundamental aspect of simplifying polynomials, as it reduces the expression to its most concise form. This simplification not only makes the expression easier to read and understand but also facilitates further operations or analysis that might be required. Let's proceed by grouping and combining the like terms to reveal the simplified form of the polynomial difference.

Combining Like Terms

After distributing the negative sign, we have the expression:

$6x^6 - x^3y^4 - 5xy^5 - 4x^5y - 2x^3y^4 - 5xy^5$

Now, let's identify and combine the like terms. We have two terms with $x^3y^4$ and two terms with $xy^5$. Combining these will simplify the expression. The $x^3y^4$ terms are $-x^3y^4$ and $-2x^3y^4$. When we combine these, we get:

$-x^3y^4 - 2x^3y^4 = -3x^3y^4$

Next, we combine the $xy^5$ terms, which are $-5xy^5$ and $-5xy^5$. This gives us:

$-5xy^5 - 5xy^5 = -10xy^5$

The terms $6x^6$ and $-4x^5y$ do not have any like terms, so they will remain unchanged. Now, let's put everything together to form the simplified polynomial. The simplified expression is crucial for determining the characteristics of the polynomial, such as its number of terms and degree. By accurately combining like terms, we ensure that our subsequent analysis is based on the most reduced and accurate form of the polynomial. This step is not just about simplifying; it's about setting the stage for a correct interpretation of the polynomial's properties.

Simplified Polynomial

After combining like terms, the simplified polynomial expression is:

$6x^6 - 4x^5y - 3x^3y^4 - 10xy^5$

This simplified form is crucial for determining the characteristics of the polynomial, such as the number of terms and the degree. Now that we have the simplified expression, we can proceed to analyze its structure. The number of terms is straightforward to count, as each term is separated by a plus or minus sign. The degree, however, requires a bit more attention. The degree of a term is the sum of the exponents of its variables, and the degree of the polynomial is the highest degree among all its terms. Understanding these concepts is essential for classifying and comparing polynomials. With our simplified polynomial in hand, we are well-equipped to determine the correct answer to the question posed. The next step involves identifying the number of terms and calculating the degree of the polynomial, which will allow us to match our findings with the given options.

Determining the Number of Terms

Now, let's count the number of terms in the simplified polynomial:

$6x^6 - 4x^5y - 3x^3y^4 - 10xy^5$

By inspection, we can see that there are four terms: $6x^6$, $-4x^5y$, $-3x^3y^4$, and $-10xy^5$. Each term is separated by either a plus or a minus sign. Accurately counting the terms is a fundamental step in characterizing a polynomial. The number of terms provides insight into the polynomial's complexity and structure. This simple count is a building block for more advanced analysis, such as determining the polynomial's degree or classifying it by type. Now that we have established the number of terms, our next objective is to determine the degree of the polynomial. This involves examining each term individually and identifying the highest degree among them. Let's proceed with this analysis to fully understand the polynomial's characteristics.

Determining the Degree

To find the degree of the polynomial, we need to determine the degree of each term and then identify the highest degree among them. The degree of a term is the sum of the exponents of its variables. Let's analyze each term in our simplified polynomial:

$6x^6$ : The degree is 6 (since the exponent of $x$ is 6). $-4x^5y$ : The degree is 5 + 1 = 6 (sum of the exponents of $x$ and $y$). $-3x^3y^4$ : The degree is 3 + 4 = 7 (sum of the exponents of $x$ and $y$). $-10xy^5$ : The degree is 1 + 5 = 6 (sum of the exponents of $x$ and $y$).

Comparing the degrees of all terms, we find that the highest degree is 7. Therefore, the degree of the polynomial is 7. Understanding the degree of a polynomial is crucial because it provides key information about the polynomial's behavior and properties. The degree influences the polynomial's graph, its end behavior, and its potential number of roots. In the context of our problem, determining the degree is essential for selecting the correct answer from the given options. Having now calculated both the number of terms and the degree, we are ready to make our final determination. Let's review our findings and match them with the answer choices provided.

The Correct Answer

We have determined that the simplified polynomial has 4 terms and a degree of 7. Now, let's look at the given options:

• A. The difference has 3 terms and a degree of 6.
• B. The difference has 4 terms and a degree of 6.
• C. The difference has 3 terms and a degree of 7.
• D. The difference has 4 terms and a degree of 7.

Comparing our findings with the options, we can see that option D matches our results. Therefore, the correct answer is:

D. The difference has 4 terms and a degree of 7.

By meticulously working through the steps of subtracting the polynomials, combining like terms, and determining the number of terms and degree, we have confidently arrived at the correct answer. This process not only solves the problem at hand but also reinforces the fundamental principles of polynomial manipulation. Understanding these principles is crucial for tackling more complex algebraic problems and for building a solid foundation in mathematics. In conclusion, careful attention to detail and a systematic approach are key to success in polynomial arithmetic.

Conclusion

In this comprehensive exploration, we successfully navigated the process of subtracting polynomials, simplifying expressions, and determining key characteristics such as the number of terms and the degree. Our journey began with the task of finding the simplified difference of two polynomials: $6x^6 - x^3y^4 - 5xy^5$ and $4x^5y + 2x^3y^4 + 5xy^5$. Through careful distribution of the negative sign and meticulous combination of like terms, we arrived at the simplified polynomial: $6x^6 - 4x^5y - 3x^3y^4 - 10xy^5$. We then systematically analyzed this simplified form, counting four distinct terms and calculating a degree of 7. This methodical approach allowed us to confidently select the correct answer from the given options: D. The difference has 4 terms and a degree of 7.

This exercise underscores the importance of a structured and detail-oriented approach to mathematical problem-solving. Each step, from the initial subtraction to the final determination of the degree, required precision and a clear understanding of the underlying principles. By breaking down the problem into manageable steps, we were able to not only arrive at the correct solution but also reinforce our understanding of polynomial operations. The concepts explored in this article, such as combining like terms and determining the degree of a polynomial, are fundamental building blocks in algebra and are essential for tackling more advanced mathematical challenges. Therefore, mastering these skills is crucial for anyone seeking to excel in mathematics and related fields.

In conclusion, the world of polynomials, while sometimes appearing intricate, can be demystified through systematic analysis and a commitment to understanding fundamental concepts. As we continue our mathematical journeys, the lessons learned from this exploration will undoubtedly serve as valuable tools in our problem-solving arsenal.