Which Is True About The Degree Of The Sum And Difference Of The Polynomials $3x^5y - 2x^3y^4 - 7xy^3$ And $8xy + 2x - Xy^3$?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

by ADMIN 235 views

Introduction

In algebra, the degree of a polynomial is a fundamental concept that helps us understand the properties and behavior of polynomial functions. When dealing with polynomials, it's essential to determine their degree, as it affects their graph, roots, and other mathematical properties. In this article, we will explore the degree of the sum and difference of two given polynomials and analyze the possible outcomes.

The Degree of a Polynomial

The degree of a polynomial is determined by the highest power of the variable (in this case, x and y) in any of its terms. For example, in the polynomial 3x5y−2x3y4−7xy33x^5y - 2x^3y^4 - 7xy^3, the highest power of x is 5, and the highest power of y is 4. Therefore, the degree of this polynomial is 5 + 4 = 9.

The Degree of the Sum and Difference of Polynomials

When adding or subtracting polynomials, we need to consider the degree of each term. The degree of the sum or difference of two polynomials is determined by the highest degree term in either polynomial.

Example Polynomials

Let's consider the two polynomials:

3x5y−2x3y4−7xy33x^5y - 2x^3y^4 - 7xy^3 ... (Polynomial 1) 8xy+2x−xy38xy + 2x - xy^3 ... (Polynomial 2)

Step 1: Identify the Highest Degree Term in Each Polynomial

In Polynomial 1, the highest degree term is 3x5y3x^5y, with a degree of 5 + 1 = 6 (since y is also a variable).

In Polynomial 2, the highest degree term is 8xy8xy, with a degree of 1 + 1 = 2.

Step 2: Determine the Degree of the Sum

To find the degree of the sum, we need to add the highest degree terms from both polynomials:

3x5y+8xy3x^5y + 8xy

The highest degree term in the sum is 3x5y3x^5y, with a degree of 5 + 1 = 6.

Step 3: Determine the Degree of the Difference

To find the degree of the difference, we need to subtract the highest degree terms from both polynomials:

3x5y−8xy3x^5y - 8xy

The highest degree term in the difference is 3x5y3x^5y, with a degree of 5 + 1 = 6.

Conclusion

Based on our analysis, we can conclude that:

  • The degree of the sum of the two polynomials is 6.
  • The degree of the difference of the two polynomials is 6.

Therefore, the correct answer is:

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

Final Thoughts

Q: What is the degree of a polynomial?

A: The degree of a polynomial is the highest power of the variable (in this case, x and y) in any of its terms. For example, in the polynomial 3x5y−2x3y4−7xy33x^5y - 2x^3y^4 - 7xy^3, the highest power of x is 5, and the highest power of y is 4. Therefore, the degree of this polynomial is 5 + 4 = 9.

Q: How do I determine the degree of a polynomial?

A: To determine the degree of a polynomial, you need to identify the highest power of the variable in any of its terms. You can do this by looking at each term and identifying the exponent of the variable. The term with the highest exponent is the one that determines the degree of the polynomial.

Q: What is the difference between the degree of a polynomial and the degree of its terms?

A: The degree of a polynomial is the highest power of the variable in any of its terms, while the degree of a term is the power of the variable in that specific term. For example, in the polynomial 3x5y−2x3y4−7xy33x^5y - 2x^3y^4 - 7xy^3, the degree of the term 3x5y3x^5y is 5 + 1 = 6, but the degree of the polynomial is 5 + 4 = 9.

Q: How do I determine the degree of the sum or difference of two polynomials?

A: To determine the degree of the sum or difference of two polynomials, you need to identify the highest degree term in either polynomial. The degree of the sum or difference is determined by the highest degree term in either polynomial.

Q: What is the degree of the sum of two polynomials?

A: The degree of the sum of two polynomials is determined by the highest degree term in either polynomial. If the highest degree terms in both polynomials have the same degree, then the degree of the sum is the same as the degree of the highest degree term. If the highest degree terms in both polynomials have different degrees, then the degree of the sum is the higher of the two degrees.

Q: What is the degree of the difference of two polynomials?

A: The degree of the difference of two polynomials is determined by the highest degree term in either polynomial. If the highest degree terms in both polynomials have the same degree, then the degree of the difference is the same as the degree of the highest degree term. If the highest degree terms in both polynomials have different degrees, then the degree of the difference is the higher of the two degrees.

Q: Can the degree of a polynomial be negative?

A: No, the degree of a polynomial cannot be negative. The degree of a polynomial is always a non-negative integer.

Q: Can the degree of a polynomial be zero?

A: Yes, the degree of a polynomial can be zero. A polynomial with a degree of zero is a constant polynomial, which is a polynomial that has no variable terms.

Q: What is the significance of the degree of a polynomial?

A: The degree of a polynomial is significant because it determines the behavior of the polynomial function. The degree of a polynomial affects its graph, roots, and other mathematical properties. For example, a polynomial with a high degree may have many roots, while a polynomial with a low degree may have few or no roots.

Q: How do I apply the concept of the degree of a polynomial in real-world problems?

A: The concept of the degree of a polynomial is applied in many real-world problems, such as:

  • Modeling population growth: A polynomial with a high degree can be used to model population growth over time.
  • Analyzing financial data: A polynomial with a low degree can be used to analyze financial data, such as stock prices or interest rates.
  • Designing electrical circuits: A polynomial with a high degree can be used to design electrical circuits, such as filters or amplifiers.

In conclusion, the degree of a polynomial is an essential concept in algebra that determines the behavior of polynomial functions. By understanding the degree of a polynomial, you can apply the concept in various real-world problems and make informed decisions.