Degree Of Sum And Difference Of Polynomials 3x^5y - 2x^3y^4 - 7xy^3 And -8x^5y + 2x^3y^4 + Xy^3

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Polynomials, fundamental building blocks in algebra, are expressions consisting of variables and coefficients, combined using addition, subtraction, and non-negative integer exponents. Understanding the degree of a polynomial is crucial for analyzing its behavior and performing operations on it. In this article, we will delve into the concept of the degree of a polynomial, specifically focusing on the sum and difference of polynomials, and explore how to determine their degrees. We will use the example polynomials 3x⁵y - 2x³y⁴ - 7xy³ and -8x⁵y + 2x³y⁴ + xy³ to illustrate the process and arrive at a definitive conclusion about their degrees.

Understanding the Degree of a Polynomial: A Deep Dive

To effectively address the question of the degree of the sum and difference of the given polynomials, it's essential to have a solid grasp of what the degree of a polynomial actually signifies. The degree of a polynomial in one variable is simply the highest power of the variable in the expression. For instance, in the polynomial x⁴ + 3x² - 2x + 1, the highest power of x is 4, so the degree of the polynomial is 4. However, when dealing with polynomials in multiple variables, like the ones in our example, the concept of degree becomes slightly more nuanced. In such cases, the degree of each term is determined by summing the exponents of all the variables in that term. The degree of the polynomial is then the highest degree among all its terms. Let's break this down with an example. Consider the term 3x⁵y. The exponent of x is 5 and the exponent of y is 1 (since y is the same as y¹). Adding these exponents gives us 5 + 1 = 6, which is the degree of this term. Now, for the polynomial 3x⁵y - 2x³y⁴ - 7xy³, we need to find the degree of each term individually:

  • 3x⁵y: Degree is 5 + 1 = 6
  • -2x³y⁴: Degree is 3 + 4 = 7
  • -7xy³: Degree is 1 + 3 = 4

The degree of the entire polynomial is the highest of these individual term degrees, which is 7. This fundamental understanding of how to determine the degree of a polynomial, especially in multiple variables, is paramount for accurately answering the question at hand. We will use this knowledge to analyze the sum and difference of the given polynomials and identify their respective degrees.

Adding and Subtracting Polynomials: A Step-by-Step Guide

Before we can determine the degree of the sum and difference of our polynomials, we need to actually perform these operations. Adding and subtracting polynomials involves combining like terms, which are terms that have the same variables raised to the same powers. This process is based on the distributive property and the concept of combining coefficients of similar terms. Let's consider our polynomials: 3x⁵y - 2x³y⁴ - 7xy³ and -8x⁵y + 2x³y⁴ + xy³. To find the sum, we simply add the corresponding terms:

Sum = (3x⁵y - 2x³y⁴ - 7xy³) + (-8x⁵y + 2x³y⁴ + xy³)

Now, we group the like terms together:

Sum = (3x⁵y - 8x⁵y) + (-2x³y⁴ + 2x³y⁴) + (-7xy³ + xy³)

Combining the coefficients of like terms, we get:

Sum = -5x⁵y + 0x³y⁴ - 6xy³

Simplifying, the sum is:

Sum = -5x⁵y - 6xy³

Next, let's find the difference between the polynomials. Subtracting polynomials requires careful attention to signs, as we need to distribute the negative sign to each term in the second polynomial:

Difference = (3x⁵y - 2x³y⁴ - 7xy³) - (-8x⁵y + 2x³y⁴ + xy³)

Distribute the negative sign:

Difference = 3x⁵y - 2x³y⁴ - 7xy³ + 8x⁵y - 2x³y⁴ - xy³

Now, group the like terms:

Difference = (3x⁵y + 8x⁵y) + (-2x³y⁴ - 2x³y⁴) + (-7xy³ - xy³)

Combine the coefficients:

Difference = 11x⁵y - 4x³y⁴ - 8xy³

Now that we have both the sum (-5x⁵y - 6xy³) and the difference (11x⁵y - 4x³y⁴ - 8xy³), we can proceed to determine their degrees, which will help us answer the original question.

Determining the Degree of the Sum: A Detailed Analysis

Having calculated the sum of the polynomials 3x⁵y - 2x³y⁴ - 7xy³ and -8x⁵y + 2x³y⁴ + xy³, we obtained the simplified expression -5x⁵y - 6xy³. Now, our focus shifts to accurately determining the degree of this resulting polynomial. Recall that the degree of a polynomial in multiple variables is the highest sum of the exponents of the variables in any of its terms. To find the degree of the sum, we need to examine each term individually and calculate the sum of its exponents.

Let's analyze the terms of the sum -5x⁵y - 6xy³:

  1. -5x⁵y: In this term, the exponent of x is 5, and the exponent of y is 1 (since y is equivalent to y¹). Adding these exponents, we get 5 + 1 = 6. Therefore, the degree of this term is 6.
  2. -6xy³: Here, the exponent of x is 1, and the exponent of y is 3. Summing these exponents gives us 1 + 3 = 4. Thus, the degree of this term is 4.

Now that we have the degrees of each term, we can determine the degree of the entire sum polynomial. The degree of the polynomial is the highest degree among its terms. In this case, we have degrees of 6 and 4. The highest of these is 6. Therefore, the degree of the sum -5x⁵y - 6xy³ is 6.

This meticulous step-by-step approach ensures that we have a clear and accurate understanding of the degree of the sum. Next, we will apply the same method to determine the degree of the difference of the polynomials.

Unveiling the Degree of the Difference: A Step-by-Step Examination

After calculating the difference between the polynomials 3x⁵y - 2x³y⁴ - 7xy³ and -8x⁵y + 2x³y⁴ + xy³, we arrived at the expression 11x⁵y - 4x³y⁴ - 8xy³. Our next crucial step is to determine the degree of this difference polynomial. Similar to how we analyzed the sum, we will examine each term in the difference individually, calculate the sum of the exponents of the variables, and then identify the highest degree among all the terms.

Let's break down the terms of the difference 11x⁵y - 4x³y⁴ - 8xy³:

  1. 11x⁵y: In this term, the exponent of x is 5, and the exponent of y is 1. Adding these exponents yields 5 + 1 = 6. Thus, the degree of this term is 6.
  2. -4x³y⁴: Here, the exponent of x is 3, and the exponent of y is 4. Summing these exponents gives us 3 + 4 = 7. Therefore, the degree of this term is 7.
  3. -8xy³: In this term, the exponent of x is 1, and the exponent of y is 3. Adding these exponents results in 1 + 3 = 4. So, the degree of this term is 4.

Now that we have the degrees of each term in the difference polynomial, we can determine the overall degree of the difference. The degree of the polynomial is the highest degree among its terms. In this case, the degrees are 6, 7, and 4. The highest of these is 7. Therefore, the degree of the difference 11x⁵y - 4x³y⁴ - 8xy³ is 7.

This methodical approach allows us to confidently identify the degree of the difference polynomial. With the degrees of both the sum and the difference now determined, we are well-equipped to answer the original question posed.

The Verdict: Sum and Difference Degrees Revealed

Having meticulously analyzed the sum and difference of the polynomials 3x⁵y - 2x³y⁴ - 7xy³ and -8x⁵y + 2x³y⁴ + xy³, we have arrived at a definitive conclusion regarding their degrees. We found that the sum, which simplifies to -5x⁵y - 6xy³, has a degree of 6. On the other hand, the difference, which is 11x⁵y - 4x³y⁴ - 8xy³, has a degree of 7.

Therefore, the correct answer is:

Both the sum and difference have a degree of 6. FALSE.

Both the sum and difference have a degree of 7. FALSE.

The sum has a degree of 6 and the difference has a degree of 7. This comprehensive analysis, breaking down each step of the process, ensures a clear understanding of how to determine the degree of the sum and difference of polynomials. From defining the degree of a polynomial to meticulously calculating the sum and difference and finally identifying their degrees, we have navigated the complexities of polynomial operations with precision. This knowledge equips us to tackle similar problems with confidence and accuracy.