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 the following:

Question

Find either the sum, difference, product, or quotient of your classmates' polynomial expressions. Choose only one operation; you do not need to perform all four.
Create rules for how to approach finding the first and last terms of your peers' polynomial expressions, without working them out fully.
Explain how you derived your rules.
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:

For your polynomial expressions, I will find the sum of the two polynomials:

Sum of \( (x^3 + 2x^2 - 3x - 4) + (x^2 + 3x - 7) \):

1. Combine like terms:
   • The highest degree term from both is \( x^3 \).
   • The \( x^2 \) terms are \( 2x^2 + x^2 = 3x^2 \).
   • The \( x \) terms are \( -3x + 3x = 0x \).
   • The constant terms are \( -4 - 7 = -11 \).

So, the sum of your polynomials is:

\[ x^3 + 3x^2 - 11 \]

Rules for Finding First and Last Terms:

1. First Term: The first term of the sum/difference of polynomials is the term with the highest degree from each polynomial being added or subtracted. Look for the terms with the largest exponent.

2. Last Term: The last term of the sum/difference will be from the constant terms of each polynomial. Simply add or subtract these constants to find what the last term will be in the result.

Derivation of Rules: These rules are derived from the definition of polynomials and how they are constructed; the polynomials are structured such that the highest degree term (first term) determines the degree of the overall polynomial, while the constant term typically indicates the value of the polynomial when \( x = 0 \) (the last term).

Limitations: These rules work effectively as long as you are combining like terms. However, if the polynomials are structured in such a way that there are no like terms for certain degrees (i.e., if one polynomial lacks a term, such as a quadratic or cubic term), the first and last terms will still apply, but you must be cautious to retain terms from both polynomials even if they don't appear in one of them.

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Response to Classmate 2:

I will find the difference of your two polynomial expressions:

Difference of \( (2x^3 + 3x^2 - 5x + 1) - (x^2 - 2x + 4) \):

1. Distribute the negative sign:

   • Change the signs for the second polynomial: \( -x^2 + 2x - 4 \).

2. Combine like terms:

   • The \( x^3 \) term is: \( 2x^3 \).
   • The \( x^2 \) terms: \( 3x^2 - x^2 = 2x^2 \).
   • The \( x \) terms: \( -5x + 2x = -3x \).
   • The constant terms: \( 1 - 4 = -3 \).

So, the difference of your polynomials is:

\[ 2x^3 + 2x^2 - 3x - 3 \]

Rules for Finding First and Last Terms:

1. First Term: Identify the term with the highest degree in the polynomial before performing the difference. This term will remain unchanged unless there is no corresponding term in the second polynomial.

2. Last Term: For the constant terms, perform the subtraction on the constants, ensuring to account for the negative sign from the second polynomial.

Derivation of Rules: The highest degree term will always dictate the polynomial’s leading term and degree, while the constant terms’ subtraction will result in the last term.

Limitations: If one polynomial is missing certain degrees (like a cubic term), it is necessary to recognize that it won’t affect the first term from the other polynomial. Moreover, the polynomials must be ordered correctly to maintain clarity while combining terms.