To solve the given problem, we need to determine the degrees of the sum and difference of the provided polynomials:
[tex]\[ \text{poly1} = 3x^5y - 2x^3y^4 - 7xy^3 \][/tex]
[tex]\[ \text{poly2} = -8x^5y + 2x^3y^4 + xy^3 \][/tex]
### Step-by-Step Solution:
1. Sum of the polynomials:
[tex]\[\text{poly1} + \text{poly2} = (3x^5y - 2x^3y^4 - 7xy^3) + (-8x^5y + 2x^3y^4 + xy^3)\][/tex]
Combine like terms:
[tex]\[ = (3x^5y - 8x^5y) + (-2x^3y^4 + 2x^3y^4) + (-7xy^3 + xy^3) \][/tex]
[tex]\[ = -5x^5y + 0 - 6xy^3 \][/tex]
[tex]\[ = -5x^5y - 6xy^3 \][/tex]
The resulting polynomial is [tex]\(-5x^5y - 6xy^3\)[/tex].
To find the degree, examine the terms:
- [tex]\(-5x^5y\)[/tex] has degree [tex]\(5+1 = 6\)[/tex]
- [tex]\(-6xy^3\)[/tex] has degree [tex]\(1+3 = 4\)[/tex]
The degree of the sum is the highest degree term, which is [tex]\(6\)[/tex].
2. Difference of the polynomials:
[tex]\[\text{poly1} - \text{poly2} = (3x^5y - 2x^3y^4 - 7xy^3) - (-8x^5y + 2x^3y^4 + xy^3)\][/tex]
Distribute the subtraction:
[tex]\[ = 3x^5y - 2x^3y^4 - 7xy^3 + 8x^5y - 2x^3y^4 - xy^3 \][/tex]
Combine like terms:
[tex]\[ = (3x^5y + 8x^5y) + (-2x^3y^4 - 2x^3y^4) + (-7xy^3 - xy^3) \][/tex]
[tex]\[ = 11x^5y - 4x^3y^4 - 8xy^3 \][/tex]
The resulting polynomial is [tex]\(11x^5y - 4x^3y^4 - 8xy^3\)[/tex].
To find the degree, examine the terms:
- [tex]\(11x^5y\)[/tex] has degree [tex]\(5+1 = 6\)[/tex]
- [tex]\(-4x^3y^4\)[/tex] has degree [tex]\(3+4 = 7\)[/tex]
- [tex]\(-8xy^3\)[/tex] has degree [tex]\(1+3 = 4\)[/tex]
The degree of the difference is the highest degree term, which is [tex]\(7\)[/tex].
### Conclusion:
The sum of the polynomials has a degree of 6, and the difference of the polynomials has a degree of 7.
Thus, the correct statement is:
- The sum has a degree of 6, but the difference has a degree of 7.
