Alright, let's look at the polynomials given:
Cory's polynomial: [tex]\( P(x) = x^7 + 3x^5 + x \)[/tex]
Melissa's polynomial: [tex]\( Q(x) = x^7 + 5x + 10 \)[/tex]
### Step-by-Step Solution
1. Adding the polynomials:
When we add the polynomials [tex]\( P(x) \)[/tex] and [tex]\( Q(x) \)[/tex]:
[tex]\[ P(x) + Q(x) = (x^7 + 3x^5 + x) + (x^7 + 5x + 10) \][/tex]
Combine like terms:
[tex]\[ (x^7 + x^7) + 3x^5 + (x + 5x) + 10 \][/tex]
[tex]\[ = 2x^7 + 3x^5 + 6x + 10 \][/tex]
The degree of the resulting polynomial is determined by the highest power of [tex]\( x \)[/tex], which is:
[tex]\[ \text{Degree of } (P(x) + Q(x)) = 7 \][/tex]
2. Subtracting the polynomials:
When we subtract polynomial [tex]\( Q(x) \)[/tex] from [tex]\( P(x) \)[/tex]:
[tex]\[ P(x) - Q(x) = (x^7 + 3x^5 + x) - (x^7 + 5x + 10) \][/tex]
Combine like terms:
[tex]\[ (x^7 - x^7) + 3x^5 + (x - 5x) - 10 \][/tex]
[tex]\[ = 0 + 3x^5 - 4x - 10 \][/tex]
[tex]\[ = 3x^5 - 4x - 10 \][/tex]
The degree of the resulting polynomial is determined by the highest power of [tex]\( x \)[/tex], which is:
[tex]\[ \text{Degree of } (P(x) - Q(x)) = 5 \][/tex]
3. Conclusion:
By comparing the degrees:
- The degree of the polynomial after addition is [tex]\(7\)[/tex].
- The degree of the polynomial after subtraction is [tex]\(5\)[/tex].
Hence, adding their polynomials together results in a polynomial with degree 7, but subtracting one polynomial from the other results in a polynomial with degree 5. Therefore, there is a difference between the degree of the sum and the degree of the difference of the polynomials.
The correct option is:
- Adding their polynomials together results in a polynomial with degree 7, but subtracting one polynomial from the other results in a polynomial with degree 5.
