Cory writes the polynomial *x*7 + 3*x*5 + 3*x* + 1. Melissa writes the polynomial *x*7 + 5*x* + 10. Is there a difference between the degree of the sum and the degree of the difference of the polynomials? 

Adding their polynomials together or subtracting one polynomial from the other both result in a polynomial with degree 7.

Adding their polynomials together or subtracting one polynomial from the other both result in a polynomial with degree 5. 

Adding their polynomials together results in a polynomial with degree 14, but subtracting one polynomial from the other results in a polynomial with degree 5.

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.

To determine if there is a difference between the degree of the sum and the degree of the difference of the polynomials written by Cory and Melissa, let's analyze the polynomials step by step.

Cory's polynomial:
\[ P(x) = x^7 + 3x^5 + 3x + 1 \]

Melissa's polynomial:
\[ Q(x) = x^7 + 5x + 10 \]

**Step 1: Adding the Polynomials**

\[ P(x) + Q(x) = (x^7 + 3x^5 + 3x + 1) + (x^7 + 5x + 10) \]
Combine like terms:
\[ P(x) + Q(x) = x^7 + x^7 + 3x^5 + 3x + 5x + 1 + 10 \]
\[ P(x) + Q(x) = 2x^7 + 3x^5 + 8x + 11 \]

The degree of \( P(x) + Q(x) \) is determined by the highest power of \( x \), which is 7.

**Step 2: Subtracting the Polynomials**

\[ P(x) - Q(x) = (x^7 + 3x^5 + 3x + 1) - (x^7 + 5x + 10) \]
Combine like terms:
\[ P(x) - Q(x) = x^7 - x^7 + 3x^5 + 3x - 5x + 1 - 10 \]
\[ P(x) - Q(x) = 3x^5 - 2x - 9 \]

The degree of \( P(x) - Q(x) \) is determined by the highest power of \( x \), which is 5.

**Conclusion:**

- The degree of the sum of the polynomials \( P(x) + Q(x) \) is 7.
- The degree of the difference of the polynomials \( P(x) - Q(x) \) is 5.

Therefore, there **is** a difference between the degree of the sum and the degree of the difference of the polynomials. 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.

Would you like more details or have any questions? Here are some questions you might find interesting:

1. How do you determine the degree of a polynomial?
2. What are the properties of polynomial addition and subtraction?
3. Can the degree of the sum of two polynomials ever be greater than the degree of the individual polynomials?
4. How do you find the leading coefficient of a polynomial?
5. What is the role of the constant term in a polynomial's degree?

**Tip:** When combining polynomials, always align like terms and combine them to simplify the expression accurately.

Understand the difference between the degree of the sum and the degree of the difference of two polynomials with detailed examples. Explore polynomial addition and subtraction with step-by-step explanations. Suitable for high school students studying advanced algebra.

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