To determine the degree of the sum and the difference of Cory's and Melissa's polynomials, let's analyze each step carefully.
First, let's write down the polynomials:
- Cory's polynomial: [tex]\(p_1(x) = x^7 + 3x^5 + 3x + 1\)[/tex]
- Melissa's polynomial: [tex]\(p_2(x) = x^7 + 5x + 10\)[/tex]
### Step-by-Step Solution:
#### 1. Sum of the Polynomials
Adding [tex]\(p_1(x)\)[/tex] and [tex]\(p_2(x)\)[/tex]:
[tex]\[ p_1(x) + p_2(x) = (x^7 + 3x^5 + 3x + 1) + (x^7 + 5x + 10) \][/tex]
Combine like terms:
[tex]\[ = (x^7 + x^7) + 3x^5 + (3x + 5x) + (1 + 10) \][/tex]
[tex]\[ = 2x^7 + 3x^5 + 8x + 11 \][/tex]
The highest power of [tex]\(x\)[/tex] in this polynomial is [tex]\(x^7\)[/tex], so the degree of the sum is 7.
#### 2. Difference of the Polynomials
Subtracting [tex]\(p_2(x)\)[/tex] from [tex]\(p_1(x)\)[/tex]:
[tex]\[ p_1(x) - p_2(x) = (x^7 + 3x^5 + 3x + 1) - (x^7 + 5x + 10) \][/tex]
Distribute the negative sign and combine like terms:
[tex]\[ = x^7 - x^7 + 3x^5 + 3x - 5x + 1 - 10 \][/tex]
[tex]\[ = 0 + 3x^5 - 2x - 9 \][/tex]
The highest power of [tex]\(x\)[/tex] in this polynomial is [tex]\(x^5\)[/tex], so the degree of the difference is 5.
#### 3. Comparing the Degrees
- The degree of the sum of the polynomials is 7.
- The degree of the difference of the polynomials is 5.
There is a difference in the degrees of the sum and the difference of the polynomials.
Conclusion: 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. Hence, there is a difference in the degrees.
