Let's analyze the given polynomials:
Polynomial 1:
[tex]\[ P_1 = 3x^5y - 2x^3y^4 - 7xy^3 \][/tex]
Polynomial 2:
[tex]\[ P_2 = -8x^5y + 2x^3y^4 + xy^3 \][/tex]
We need to find the degree of the sum of the polynomials and the degree of the difference of the polynomials.
### Sum of the Polynomials:
The sum of [tex]\( P_1 \)[/tex] and [tex]\( P_2 \)[/tex] is:
[tex]\[ P_1 + P_2 = (3x^5y - 2x^3y^4 - 7xy^3) + (-8x^5y + 2x^3y^4 + xy^3) \][/tex]
Combining like terms:
[tex]\[ P_1 + P_2 = (3x^5y - 8x^5y) + (-2x^3y^4 + 2x^3y^4) + (-7xy^3 + xy^3) \][/tex]
[tex]\[ = -5x^5y + 0x^3y^4 - 6xy^3 \][/tex]
[tex]\[ = -5x^5y - 6xy^3 \][/tex]
Now, we need to find the degree of the sum. The highest degree term in the sum is [tex]\( -5x^5y \)[/tex], which has a degree calculated as [tex]\( 5 + 1 = 6 \)[/tex].
### Difference of the Polynomials:
The difference of [tex]\( P_1 \)[/tex] and [tex]\( P_2 \)[/tex] is:
[tex]\[ P_1 - P_2 = (3x^5y - 2x^3y^4 - 7xy^3) - (-8x^5y + 2x^3y^4 + xy^3) \][/tex]
Distributing the negative sign:
[tex]\[ P_1 - P_2 = (3x^5y - 2x^3y^4 - 7xy^3) + (8x^5y - 2x^3y^4 - xy^3) \][/tex]
Combining like terms:
[tex]\[ P_1 - P_2 = (3x^5y + 8x^5y) + (-2x^3y^4 - 2x^3y^4) + (-7xy^3 - xy^3) \][/tex]
[tex]\[ = 11x^5y - 4x^3y^4 - 8xy^3 \][/tex]
Now, we need to find the degree of the difference. The highest degree term in the difference is [tex]\( -4x^3y^4 \)[/tex], which has a degree calculated as [tex]\( 3 + 4 = 7 \)[/tex].
### Conclusion:
The sum [tex]\( P_1 + P_2 \)[/tex] has a degree of 6.
The difference [tex]\( P_1 - P_2 \)[/tex] has a degree of 7.
Therefore, the correct statement is:
The sum has a degree of 6, but the difference has a degree of 7.
