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. I will find the sum of your polynomial expressions: \[ (x^3 + 2x^2 - 3x - 4) + (x^2 + 3x - 7) \] Combine like terms: \[ = x^3 + (2x^2 + x^2) + (-3x + 3x) + (-4 - 7) \] \[ = x^3 + 3x^2 - 11 \] The resulting polynomial is \(x^3 + 3x^2 - 11\) with a degree of 3.

2. Rules for first and last terms:

   • Finding the first term: Look for the term with the highest power of \(x\). This is found by identifying the term with the highest exponent.
   • Finding the last term: Look for the constant term, which is typically the term with no \(x\) present. This is usually the term that stands alone.

3. Explanation of rules: The highest power term (first term) defines the leading term which determines the polynomial's end behavior and degree. The constant term (last term) provides the polynomial's value when \(x = 0\).

4. Limitations of these rules: These rules are applicable only to polynomials where the terms are arranged in standard form (from highest degree to lowest). They may not work as expected if polynomial expressions are not simplified or arranged correctly.

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Response to Classmate 2:

1. I will explore the difference of your polynomial expressions: \[ (2x^3 + 3x^2 - 5x + 1) - (x^2 - 2x + 4) \] Distributing the negative: \[ = 2x^3 + 3x^2 - 5x + 1 - x^2 + 2x - 4 \] Now, combine like terms: \[ = 2x^3 + (3x^2 - x^2) + (-5x + 2x) + (1 - 4) \] \[ = 2x^3 + 2x^2 - 3x - 3 \] The resulting polynomial is \(2x^3 + 2x^2 - 3x - 3\) with a degree of 3.

2. Rules for first and last terms:

   • Finding the first term: Identify the term with the highest exponent from the polynomial, which represents its degree.
   • Finding the last term: The constant term at the end of the polynomial represents the polynomial's value when evaluating \(x = 0\).

3. Explanation of rules: Identifying the first term is straightforward since it highlights the overall shape and growth of the polynomial, while the last term indicates the starting point of the polynomial on the y-axis.

4. Limitations of these rules: The rules assume that the polynomial is simplified and follows standard form. If polynomials are not arranged correctly, these rules could lead to incorrect identifications of terms. Additionally, they may not apply to non-polynomial expressions.