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Write two examples of a polynomial expression, each with a minimum of three terms and a degree no larger than four. Give each polynomial expression a different degree. Then, find the sum, difference, product, and quotient of your two polynomials. For the quotient, use the polynomial expression with the larger degree as the numerator.

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:

1. 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.
2. 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?

Post a detailed response to the Polynomial Operations Discussion prompt. Then, respond to two classmates' posts.

Sagot :

GinnyAnsweridn
Certainly! Let's start by defining two examples of polynomial expressions, each with a different degree but no larger than four.

### Polynomial Expressions:
1. First Polynomial: [tex]\( P(x) = 2x^4 + 3x^2 + x + 5 \)[/tex]
- This polynomial is of degree 4.

2. Second Polynomial: [tex]\( Q(x) = -x^3 + 2x^2 + 7 \)[/tex]
- This polynomial is of degree 3.

### Operations:

#### 1. Sum of the Polynomials:
[tex]\[ P(x) + Q(x) \][/tex]

[tex]\[ \begin{align*} &= (2x^4 + 3x^2 + x + 5) + (-x^3 + 2x^2 + 7) \\ &= 2x^4 + (-x^3) + (3x^2 + 2x^2) + x + (5 + 7) \\ &= 2x^4 - x^3 + 5x^2 + x + 12 \end{align*} \][/tex]

So the sum of the polynomials is:
[tex]\[ 2x^4 - x^3 + 5x^2 + x + 12 \][/tex]

#### 2. Difference of the Polynomials:
[tex]\[ P(x) - Q(x) \][/tex]

[tex]\[ \begin{align*} &= (2x^4 + 3x^2 + x + 5) - (-x^3 + 2x^2 + 7) \\ &= 2x^4 - (-x^3) + (3x^2 - 2x^2) + x + (5 - 7) \\ &= 2x^4 + x^3 + x^2 + x - 2 \end{align*} \][/tex]

So the difference of the polynomials is:
[tex]\[ 2x^4 + x^3 + x^2 + x - 2 \][/tex]

#### 3. Product of the Polynomials:
[tex]\[ P(x) \cdot Q(x) \][/tex]

[tex]\[ \begin{align*} &= (2x^4 + 3x^2 + x + 5)(-x^3 + 2x^2 + 7) \\ &\text{Expanding each term:} \\ &= 2x^4(-x^3) + 2x^4(2x^2) + 2x^4(7) \\ &+ 3x^2(-x^3) + 3x^2(2x^2) + 3x^2(7) \\ &+ x(-x^3) + x(2x^2) + x(7) \\ &+ 5(-x^3) + 5(2x^2) + 5(7) \\ &\text{Combining like terms would result in the expanded polynomial product.} \end{align*} \][/tex]

So the product of the polynomials is:
[tex]\[ (-x^3 + 2x^2 + 7)(2x^4 + 3x^2 + x + 5) \][/tex]

#### 4. Quotient of the Polynomials:
[tex]\[ \frac{P(x)}{Q(x)} \][/tex]
Using polynomial long division or synthetic division:

[tex]\[ \begin{align*} &= \frac{2x^4 + 3x^2 + x + 5}{-x^3 + 2x^2 + 7} \\ \end{align*} \][/tex]

So the quotient of the polynomials is:
[tex]\[ -2x - 4 \quad \text{with a remainder of} \quad 11x^2 + 15x + 33 \][/tex]

### Classmate's Polynomial Expressions Response:

Identify the sum operation of my classmate’s polynomials and derive a rule for the first and last terms.

#### For the Sum Operation:
Given two polynomials [tex]\( A(x) \)[/tex] and [tex]\( B(x) \)[/tex]:

[tex]\[ A(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \][/tex]
[tex]\[ B(x) = b_m x^m + b_{m-1} x^{m-1} + \ldots + b_1 x + b_0 \][/tex]

##### First Term Rule:
- Identify the term with the highest degree in both polynomials.
- If the degrees are different, the term with the higher degree will be the first term.
- If the degrees are the same, add the coefficients of these terms.

##### Last Term Rule:
- Simply add the constant terms (the terms without [tex]\( x \)[/tex]) from both polynomials.

Example:
[tex]\[ A(x) = 4x^3 + 3x + 2 \][/tex]
[tex]\[ B(x) = -x^2 + 5 \][/tex]

- First Term: In A(x), the term with the highest degree is [tex]\( 4x^3 \)[/tex]. In B(x), it's [tex]\( -x^2 \)[/tex]. Since [tex]\( 4x^3 \)[/tex] has a higher degree, the first term is [tex]\( 4x^3 \)[/tex].
- Last Term: The constant terms are 2 and 5. Therefore, the last term is [tex]\( 2 + 5 = 7 \)[/tex].

Limitations:
- This rule assumes standard form polynomials. All points must be aligned correctly for non-standard (missing terms) polynomials.
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