To determine the degree of the sum and difference of the given polynomials, we start by examining the polynomials themselves:
First polynomial:
[tex]\[ 3x^5 y - 2x^3 y^4 - 7xy^3 \][/tex]
Second polynomial:
[tex]\[ -8x^5 y + 2x^3 y^4 + xy^3 \][/tex]
### Step 1: Sum of the Polynomials
We add the corresponding terms of the two polynomials:
[tex]\[ (3x^5 y - 2x^3 y^4 - 7xy^3) + (-8x^5 y + 2x^3 y^4 + xy^3) \][/tex]
Combine like terms:
[tex]\[ (3x^5y - 8x^5y) + (-2x^3y^4 + 2x^3y^4) + (-7xy^3 + xy^3) \][/tex]
[tex]\[ -5x^5y + 0x^3y^4 - 6xy^3 \][/tex]
The simplified sum polynomial is:
[tex]\[ -5x^5 y - 6xy^3 \][/tex]
The degree of a polynomial is determined by the term with the highest sum of exponents (the degree of each term is found by adding the exponents of its variables).
- [tex]\( -5x^5 y \)[/tex] has a degree of [tex]\(5 + 1 = 6\)[/tex]
- [tex]\( -6xy^3 \)[/tex] has a degree of [tex]\(1 + 3 = 4\)[/tex]
The highest degree is 6, so the sum of the polynomials has a degree of 6.
### Step 2: Difference of the Polynomials
We subtract the second polynomial from the first:
[tex]\[ (3x^5 y - 2x^3 y^4 - 7xy^3) - (-8x^5 y + 2x^3 y^4 + xy^3) \][/tex]
Distribute the negative sign:
[tex]\[ 3x^5 y - 2x^3 y^4 - 7xy^3 + 8x^5 y - 2x^3 y^4 - xy^3 \][/tex]
Combine like terms:
[tex]\[ (3x^5 y + 8x^5 y) + (-2x^3 y^4 - 2x^3 y^4) + (-7xy^3 - xy^3) \][/tex]
[tex]\[ 11x^5 y - 4x^3 y^4 - 8xy^3 \][/tex]
The simplified difference polynomial is:
[tex]\[ 11x^5 y - 4x^3 y^4 - 8xy^3 \][/tex]
Now, check the degree of each term:
- [tex]\( 11x^5 y \)[/tex] has a degree of [tex]\(5 + 1 = 6\)[/tex]
- [tex]\( -4x^3 y^4 \)[/tex] has a degree of [tex]\(3 + 4 = 7\)[/tex]
- [tex]\( -8xy^3 \)[/tex] has a degree of [tex]\(1 + 3 = 4\)[/tex]
The highest degree is 7, so the difference of the polynomials has a degree of 7.
### Conclusion
From the steps above, we have determined:
- The degree of the sum of the polynomials is 6.
- The degree of the difference of the polynomials is 7.
Thus, the correct statement is:
"The sum has a degree of 6, but the difference has a degree of 7."
