After you have posted your polynomial expressions and their sum, difference, product, and quotient, respond to two of your classmates' posts. In your response to each post, complete all the questions for each classmate:

Question

1. Find either the sum, difference, product, or quotient of your classmates' polynomial expressions.
2. Choose only one operation; you do not need to perform all four.
3. Create rules for how to approach finding the first and last terms of your peers' polynomial expressions, without working them out fully.
4. Explain how you derived your rules.
5. Do any of these rules have limitations?

Classmate 1: X^3+2^2-3x-4
degree is 3
x^2+3x-7
degree is 2
sum(x^2+2x^2-3x-4)+(x^2+3x-7)=^4+5x^2-10x-11
degree is 4
Difference:(x^2+2x^2-3x-4)-(x^2+3x-7)=x^2-x-10
degree is 2

Classmate 2: Polynomial A(2x^3 + 3x^2 - 5X + 1) + Polynomial B(x^2 - 2x + 4) Sum 2x^3 + 4x^2 - 7x +5

(2x^3 + 3x^2 - 5X + 1) - (x^2 - 2x + 4) Difference 2x^3 + 2x^2 - 3x - 3

(2x^3 + 3x^2 - 5X + 1) (x^2 - 2x + 4) Product 2x^5 - x^4 + 8x^3 - 11x^2 + 18x + 4

Quotient 2x^3 + 3x^2 - 5x + 1 2x + 7 + - 14x - 27
x^2 - 2x + 4 x^2 - 2x +4

Answer 1

Response to Classmate 1

1. Find the Difference: To find the difference of your polynomial expressions: \[ (x^3 + 2x^2 - 3x - 4) - (x^2 + 3x - 7) \] Simplifying this: \[ x^3 + 2x^2 - 3x - 4 - x^2 - 3x + 7 = x^3 + (2x^2 - x^2) + (-3x - 3x) + (-4 + 7) \] This results in: \[ x^3 + x^2 - 6x + 3 \]

2. Rules for Finding Terms:

   • First Term: The first term of the difference will be the term from the polynomial with the highest degree (x^3 in this case), as long as it is positive. If it's negative after simplifying, it will be considered.
   • Last Term: The last term results from taking the constant terms of both polynomials and subtracting them, noting that you may need to combine them.

3. Explanation of Rules: These rules are based on the nature of polynomial arithmetic. The first term comes from the highest degree polynomial, so you look at the leading coefficients after performing the operation. The last term is found from constant values, requiring careful attention to signs when combining.

4. Limitations of the Rules: One limitation is that when polynomials have different leading terms or are structured differently (like varying degrees), more checks are necessary to ensure all terms are accounted for appropriately.

---

Response to Classmate 2

1. Finding the Sum: Let's confirm the sum you provided: \[ (2x^3 + 3x^2 - 5x + 1) + (x^2 - 2x + 4) \] Simplifying: \[ 2x^3 + (3x^2 + x^2) + (-5x - 2x) + (1 + 4) = 2x^3 + 4x^2 - 7x + 5 \] It looks like your sum is correct!

2. Rules for Finding Terms:

   • First Term: The first term in the sum is simply the term with the highest degree from the two polynomials, which is the leading term (2x^3).
   • Last Term: The last term comes from combining the constant terms of the polynomials (1 + 4).

3. Explanation of Rules: These rules follow the principle that when adding polynomials, you maintain the degree of leading terms and combine corresponding coefficients for lower degree terms. This method helps streamline simplifications while ensuring you retain all necessary terms.

4. Limitations of the Rules: One limitation is during multiplication and division, where the degree and leading coefficients might change significantly, thus altering the straightforward application of the first and last term rules, requiring more extensive computation and verification for each operation.