Let's analyze the problem step by step:
1. Identify the given polynomials:
- Cory's polynomial: [tex]\( x^7 + 3x^5 + 3x + 1 \)[/tex]
- Melissa's polynomial: [tex]\( x^7 + 5x + 10 \)[/tex]
2. Sum of the polynomials:
Adding the two polynomials:
[tex]\[
(x^7 + 3x^5 + 3x + 1) + (x^7 + 5x + 10)
\][/tex]
Combine like terms:
[tex]\[
x^7 + x^7 + 3x^5 + 3x + 5x + 1 + 10 = 2x^7 + 3x^5 + 8x + 11
\][/tex]
The degree of the resulting polynomial is determined by the highest power of [tex]\( x \)[/tex]. In this case, the highest power is [tex]\( x^7 \)[/tex].
Therefore, the degree of the sum is 7.
3. Difference of the polynomials:
Subtracting Melissa's polynomial from Cory's polynomial:
[tex]\[
(x^7 + 3x^5 + 3x + 1) - (x^7 + 5x + 10)
\][/tex]
Combine like terms:
[tex]\[
x^7 - x^7 + 3x^5 + 3x - 5x + 1 - 10 = 3x^5 - 2x - 9
\][/tex]
The degree of the resulting polynomial is determined by the highest power of [tex]\( x \)[/tex]. In this case, the highest power is [tex]\( x^5 \)[/tex].
Therefore, the degree of the difference is 5.
4. Comparison of the degrees:
- Degree of the sum: 7
- Degree of the difference: 5
From the analysis, we see that adding their polynomials results in a polynomial with degree 7, while subtracting one polynomial from the other results in a polynomial with degree 5.
Therefore, the correct statement 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.
