Answer :
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.
### 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.