Degree Of Sum And Difference Of Polynomials 3x⁵y – 2x³y⁴ – 7xy³ And –8x⁵y + 2x³y⁴ + Xy³

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When dealing with polynomials, understanding their degrees is crucial for various algebraic operations and analyses. The degree of a polynomial is the highest sum of the exponents of variables in any term within the polynomial. To find the degree of the sum and difference of two polynomials, we must first perform the operations and then identify the term with the highest degree. This article aims to provide a comprehensive explanation of how to determine the degrees of the sum and difference of the given polynomials: 3x⁵y – 2x³y⁴ – 7xy³ and –8x⁵y + 2x³y⁴ + xy³.

Understanding Polynomial Degrees

Before diving into the specific problem, it's essential to understand the concept of the degree of a polynomial. The degree of a term in a polynomial is the sum of the exponents of its variables. For example, in the term 3x⁵y, the degree is 5 + 1 = 6. In the term –2x³y⁴, the degree is 3 + 4 = 7. The degree of the polynomial itself is the highest degree among all its terms. For the polynomial 3x⁵y – 2x³y⁴ – 7xy³, the degrees of the terms are 6, 7, and 4 respectively, making the degree of the polynomial 7.

When adding or subtracting polynomials, terms with the same variables and exponents (like terms) are combined. The degree of the resulting polynomial is determined by the highest degree among the terms that remain after the combination. If terms of the highest degree cancel each other out, then the degree of the resulting polynomial will be lower. Understanding these principles is vital for accurately determining the degrees of the sum and difference of polynomials.

Calculating the Sum of the Polynomials

To find the sum of the polynomials 3x⁵y – 2x³y⁴ – 7xy³ and –8x⁵y + 2x³y⁴ + xy³, we add the like terms together. This involves combining terms with the same variables raised to the same powers. Let's break down the process step by step:

  1. Write down the two polynomials:
    • Polynomial 1: 3x⁵y – 2x³y⁴ – 7xy³
    • Polynomial 2: –8x⁵y + 2x³y⁴ + xy³
  2. Add the polynomials by combining like terms:
    • (3x⁵y – 2x³y⁴ – 7xy³) + (–8x⁵y + 2x³y⁴ + xy³)
  3. Combine the x⁵y terms: 3x⁵y + (-8x⁵y) = -5x⁵y
  4. Combine the x³y⁴ terms: -2x³y⁴ + 2x³y⁴ = 0 (These terms cancel each other out)
  5. Combine the xy³ terms: -7xy³ + xy³ = -6xy³
  6. Write the resulting polynomial: -5x⁵y - 6xy³

The resulting polynomial after addition is -5x⁵y - 6xy³. To find the degree of this sum, we identify the degrees of each term:

  • The degree of -5x⁵y is 5 + 1 = 6.
  • The degree of -6xy³ is 1 + 3 = 4.

The highest degree among these terms is 6. Therefore, the degree of the sum of the polynomials is 6. The process of adding polynomials involves identifying and combining like terms, which are terms that have the same variables raised to the same powers. Careful attention to signs and exponents is essential to ensure the accuracy of the resulting polynomial. In this case, the cancellation of the x³y⁴ terms significantly impacts the final degree of the sum, highlighting the importance of thorough calculation.

Calculating the Difference of the Polynomials

Next, we calculate the difference between the polynomials 3x⁵y – 2x³y⁴ – 7xy³ and –8x⁵y + 2x³y⁴ + xy³. Subtracting polynomials involves changing the sign of each term in the second polynomial and then adding the like terms. This process ensures that we correctly account for the subtraction operation.

  1. Write down the two polynomials:
    • Polynomial 1: 3x⁵y – 2x³y⁴ – 7xy³
    • Polynomial 2: –8x⁵y + 2x³y⁴ + xy³
  2. Subtract the polynomials by changing the signs of the second polynomial and adding:
    • (3x⁵y – 2x³y⁴ – 7xy³) – (–8x⁵y + 2x³y⁴ + xy³)
    • = 3x⁵y – 2x³y⁴ – 7xy³ + 8x⁵y – 2x³y⁴ – xy³
  3. Combine the x⁵y terms: 3x⁵y + 8x⁵y = 11x⁵y
  4. Combine the x³y⁴ terms: -2x³y⁴ - 2x³y⁴ = -4x³y⁴
  5. Combine the xy³ terms: -7xy³ - xy³ = -8xy³
  6. Write the resulting polynomial: 11x⁵y – 4x³y⁴ – 8xy³

The resulting polynomial after subtraction is 11x⁵y – 4x³y⁴ – 8xy³. Now, we determine the degree of this difference by identifying the degrees of each term:

  • The degree of 11x⁵y is 5 + 1 = 6.
  • The degree of -4x³y⁴ is 3 + 4 = 7.
  • The degree of -8xy³ is 1 + 3 = 4.

The highest degree among these terms is 7. Therefore, the degree of the difference of the polynomials is 7. The subtraction process requires careful distribution of the negative sign across all terms of the second polynomial. Accurately combining like terms after this distribution is crucial for obtaining the correct resulting polynomial. The highest degree among the terms in the final polynomial dictates the degree of the difference.

Determining the Correct Answer

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

  • The sum of the polynomials 3x⁵y – 2x³y⁴ – 7xy³ and –8x⁵y + 2x³y⁴ + xy³ is -5x⁵y - 6xy³, which has a degree of 6.
  • The difference of the polynomials is 11x⁵y – 4x³y⁴ – 8xy³, which has a degree of 7.

Now, let's evaluate the given options:

  • (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.

Comparing our results with the options, we can see that option (C) accurately reflects our findings. The sum has a degree of 6, and the difference has a degree of 7. Therefore, the correct answer is (C).

Conclusion

In summary, determining the degree of the sum and difference of polynomials involves performing the respective operations and then identifying the highest degree among the resulting terms. For the given polynomials 3x⁵y – 2x³y⁴ – 7xy³ and –8x⁵y + 2x³y⁴ + xy³, the sum has a degree of 6, and the difference has a degree of 7. Thus, the correct answer is (C). Understanding the fundamental concepts of polynomial degrees and accurately executing addition and subtraction operations are essential for solving such problems. This exercise highlights the importance of careful algebraic manipulation and attention to detail in polynomial arithmetic. The ability to determine polynomial degrees is a crucial skill in algebra, with applications in various mathematical and scientific contexts. Mastering these concepts provides a solid foundation for more advanced topics in mathematics.