Which Is True About The Degree Of The Sum And Difference Of The Polynomials $3x^5y - 2x^3y^4 - 7xy^3$ And $-8x^5y + 2x^9y^4 + Xy^3$?A. Both The Sum And Difference Have A Degree Of 6.B. Both The Sum And Difference Have A Degree Of 7.C.

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 characteristics. 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 Sum and Difference of Polynomials

When adding or subtracting polynomials, we need to combine like terms, which are terms with the same variable and exponent. The degree of the resulting polynomial will be determined by the highest degree term in the sum or difference.

Analyzing the Given Polynomials

Let's consider the two given polynomials:

  1. 3x5y−2x3y4−7xy33x^5y - 2x^3y^4 - 7xy^3
  2. −8x5y+2x9y4+xy3-8x^5y + 2x^9y^4 + xy^3

To find the degree of the sum and difference of these polynomials, we need to identify the highest degree term in each polynomial.

Highest Degree Terms

In the first polynomial, the highest degree term is 3x5y3x^5y, with a degree of 5 + 1 = 6 (since y is considered to have a degree of 1).

In the second polynomial, the highest degree term is 2x9y42x^9y^4, with a degree of 9 + 4 = 13.

The Sum of the Polynomials

When adding the two polynomials, we get:

3x5y−2x3y4−7xy3−8x5y+2x9y4+xy33x^5y - 2x^3y^4 - 7xy^3 - 8x^5y + 2x^9y^4 + xy^3

Combining like terms, we get:

−5x5y−2x3y4+2x9y4−6xy3-5x^5y - 2x^3y^4 + 2x^9y^4 - 6xy^3

The highest degree term in the sum is 2x9y42x^9y^4, with a degree of 9 + 4 = 13.

The Difference of the Polynomials

When subtracting the second polynomial from the first, we get:

3x5y−2x3y4−7xy3−(−8x5y+2x9y4+xy3)3x^5y - 2x^3y^4 - 7xy^3 - (-8x^5y + 2x^9y^4 + xy^3)

Simplifying, we get:

11x5y−2x3y4−7xy3−2x9y4−xy311x^5y - 2x^3y^4 - 7xy^3 - 2x^9y^4 - xy^3

Combining like terms, we get:

11x5y−2x9y4−8x3y4−8xy311x^5y - 2x^9y^4 - 8x^3y^4 - 8xy^3

The highest degree term in the difference is −2x9y4-2x^9y^4, with a degree of 9 + 4 = 13.

Conclusion

Based on our analysis, we can conclude that:

  • The degree of the sum of the two polynomials is 13.
  • The degree of the difference of the two polynomials is also 13.

Therefore, the correct answer is:

C. Both the sum and difference have a degree of 13.

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 the exponents of the variable in each term and finding the highest one.

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

A: The degree of a polynomial is the highest power of the variable in any of its terms, while the order of a polynomial is the number of terms it contains. For example, the polynomial x2+2x+1x^2 + 2x + 1 has a degree of 2 (since the highest power of x is 2) and an order of 3 (since it contains three terms).

Q: Can a polynomial have a negative degree?

A: No, a polynomial cannot have a negative degree. The degree of a polynomial is always a non-negative integer, since it represents the highest power of the variable in any of its terms.

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

A: To find the degree of the sum or difference of two polynomials, you need to combine like terms and identify the highest degree term in the resulting polynomial. The degree of the sum or difference will be the same as the degree of the highest degree term.

Q: Can the degree of a polynomial change when it is multiplied by a constant?

A: No, the degree of a polynomial does not change when it is multiplied by a constant. Multiplying a polynomial by a constant only changes the coefficients of its terms, but not the degree.

Q: How do I use the degree of a polynomial to solve problems?

A: The degree of a polynomial can be used to solve problems in various ways, such as:

  • Determining the number of real roots of a polynomial
  • Finding the x-intercepts of a polynomial
  • Identifying the behavior of a polynomial as x approaches positive or negative infinity
  • Solving systems of equations involving polynomials

Q: What are some common applications of the degree of a polynomial?

A: The degree of a polynomial has many applications in various fields, such as:

  • Algebra: The degree of a polynomial is used to determine the number of real roots and the behavior of the polynomial.
  • Calculus: The degree of a polynomial is used to find the x-intercepts and the behavior of the polynomial as x approaches positive or negative infinity.
  • Engineering: The degree of a polynomial is used to design and analyze systems, such as electrical circuits and mechanical systems.
  • Computer Science: The degree of a polynomial is used in algorithms and data structures, such as polynomial time algorithms and polynomial space complexity.

Conclusion

In conclusion, the degree of a polynomial is a fundamental concept in algebra that has many applications in various fields. By understanding the degree of a polynomial, you can solve problems and make informed decisions in various mathematical and real-world applications.