Let's break down the problem step by step to understand the degrees of the sums and differences of the given polynomials.
1. Definition of Polynomials:
- Richard's polynomial: \( p_1(x) = x^7 + 3x^5 + 3x + 1 \)
- Melissa's polynomial: \( p_2(x) = x^7 + 5x + 10 \)
2. Adding the Polynomials:
To find the sum of these polynomials, we add the corresponding coefficients:
[tex]\[
p_1(x) + p_2(x) = (x^7 + 3x^5 + 3x + 1) + (x^7 + 5x + 10)
\][/tex]
Combining like terms, we get:
[tex]\[
x^7 + x^7 + 3x^5 + 3x + 5x + 1 + 10 = 2x^7 + 3x^5 + 8x + 11
\][/tex]
The highest degree term in the resulting polynomial is \(2x^7\), so the degree of the sum is \(7\).
3. Subtracting the Polynomials:
To find the difference of these polynomials, we subtract the corresponding coefficients:
[tex]\[
p_1(x) - p_2(x) = (x^7 + 3x^5 + 3x + 1) - (x^7 + 5x + 10)
\][/tex]
Combining like terms, we get:
[tex]\[
x^7 - x^7 + 3x^5 + 3x - 5x + 1 - 10 = 3x^5 - 2x - 9
\][/tex]
The highest degree term in the resulting polynomial is \(3x^5\), so the degree of the difference is \(5\).
Hence, 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.
So, the correct answer 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.
