Answer :
Certainly! Let's approach this problem step-by-step and address each part of the question in detail.
### Polynomial Expressions and Division
Step 1: Define Two Polynomial Expressions
Consider the following two polynomial expressions:
- Numerator: [tex]\( x^2 - 4x + 4 \)[/tex]
- Denominator: [tex]\( x - 2 \)[/tex]
We will perform polynomial division to find the quotient and remainder.
Step 2: Perform Polynomial Division
To divide [tex]\( x^2 - 4x + 4 \)[/tex] by [tex]\( x - 2 \)[/tex]:
1. Divide the leading term of the numerator by the leading term of the denominator:
[tex]\[ \frac{x^2}{x} = x \][/tex]
2. Multiply the entire denominator by this quotient term and subtract from the numerator:
[tex]\[ x^2 - 4x + 4 - (x \cdot (x - 2)) = x^2 - 4x + 4 - (x^2 - 2x) = -2x + 4 \][/tex]
3. Repeat the process with the new polynomial [tex]\(-2x + 4\)[/tex]:
[tex]\[ \frac{-2x}{x} = -2 \][/tex]
4. Multiply the entire denominator by the new quotient term and subtract:
[tex]\[ -2x + 4 - (-2 \cdot (x - 2)) = -2x + 4 - (-2x + 4) = 0 \][/tex]
Thus, the quotient is [tex]\( x - 2 \)[/tex] and the remainder is [tex]\( 0 \)[/tex].
### Check for Closure Under Division
A set is closed under an operation if performing that operation on elements within the set always results in another element within the set. In our case:
- The quotient [tex]\( x - 2 \)[/tex] is a polynomial.
- The remainder [tex]\( 0 \)[/tex] is also a polynomial.
However, polynomial division does not always produce a polynomial quotient unless the remainder is zero.
Conclusion: Polynomials are not closed under division because the quotient of two polynomials is not necessarily a polynomial (it can be a rational function when the remainder is not zero).
### Comparing Addition, Subtraction, and Multiplication with Division
Addition:
- Adding two polynomials [tex]\( (a(x) + b(x)) \)[/tex] results in another polynomial.
- Example: [tex]\( (x^2 + 3x + 2) + (x - 1) = x^2 + 4x + 1 \)[/tex]
Subtraction:
- Subtracting one polynomial from another [tex]\( (a(x) - b(x)) \)[/tex] also results in another polynomial.
- Example: [tex]\( (x^2 + 3x + 2) - (x - 1) = x^2 + 2x + 3 \)[/tex]
Multiplication:
- Multiplying two polynomials [tex]\( (a(x) \cdot b(x)) \)[/tex] results in another polynomial.
- Example: [tex]\( (x^2 + 3x + 2) \cdot (x - 1) = x^3 + 2x^2 - x - 2 \)[/tex]
Division:
- Division of polynomials, as previously mentioned, can result in a rational function if the remainder is not zero.
- Example: [tex]\( \frac{x^2 - 2}{x - 2} \neq \text{Polynomial} \)[/tex] (since it results in [tex]\( x + 2 + \frac{2}{x-2} \)[/tex]).
### Summary
- Addition, Subtraction, and Multiplication: These operations on polynomials always result in another polynomial. Hence, polynomials are closed under these operations.
- Division: This operation can yield rational functions, not necessarily polynomials. Therefore, polynomials are not closed under division.
I hope this detailed explanation helps in understanding the nature of polynomial operations and their closures!
### Polynomial Expressions and Division
Step 1: Define Two Polynomial Expressions
Consider the following two polynomial expressions:
- Numerator: [tex]\( x^2 - 4x + 4 \)[/tex]
- Denominator: [tex]\( x - 2 \)[/tex]
We will perform polynomial division to find the quotient and remainder.
Step 2: Perform Polynomial Division
To divide [tex]\( x^2 - 4x + 4 \)[/tex] by [tex]\( x - 2 \)[/tex]:
1. Divide the leading term of the numerator by the leading term of the denominator:
[tex]\[ \frac{x^2}{x} = x \][/tex]
2. Multiply the entire denominator by this quotient term and subtract from the numerator:
[tex]\[ x^2 - 4x + 4 - (x \cdot (x - 2)) = x^2 - 4x + 4 - (x^2 - 2x) = -2x + 4 \][/tex]
3. Repeat the process with the new polynomial [tex]\(-2x + 4\)[/tex]:
[tex]\[ \frac{-2x}{x} = -2 \][/tex]
4. Multiply the entire denominator by the new quotient term and subtract:
[tex]\[ -2x + 4 - (-2 \cdot (x - 2)) = -2x + 4 - (-2x + 4) = 0 \][/tex]
Thus, the quotient is [tex]\( x - 2 \)[/tex] and the remainder is [tex]\( 0 \)[/tex].
### Check for Closure Under Division
A set is closed under an operation if performing that operation on elements within the set always results in another element within the set. In our case:
- The quotient [tex]\( x - 2 \)[/tex] is a polynomial.
- The remainder [tex]\( 0 \)[/tex] is also a polynomial.
However, polynomial division does not always produce a polynomial quotient unless the remainder is zero.
Conclusion: Polynomials are not closed under division because the quotient of two polynomials is not necessarily a polynomial (it can be a rational function when the remainder is not zero).
### Comparing Addition, Subtraction, and Multiplication with Division
Addition:
- Adding two polynomials [tex]\( (a(x) + b(x)) \)[/tex] results in another polynomial.
- Example: [tex]\( (x^2 + 3x + 2) + (x - 1) = x^2 + 4x + 1 \)[/tex]
Subtraction:
- Subtracting one polynomial from another [tex]\( (a(x) - b(x)) \)[/tex] also results in another polynomial.
- Example: [tex]\( (x^2 + 3x + 2) - (x - 1) = x^2 + 2x + 3 \)[/tex]
Multiplication:
- Multiplying two polynomials [tex]\( (a(x) \cdot b(x)) \)[/tex] results in another polynomial.
- Example: [tex]\( (x^2 + 3x + 2) \cdot (x - 1) = x^3 + 2x^2 - x - 2 \)[/tex]
Division:
- Division of polynomials, as previously mentioned, can result in a rational function if the remainder is not zero.
- Example: [tex]\( \frac{x^2 - 2}{x - 2} \neq \text{Polynomial} \)[/tex] (since it results in [tex]\( x + 2 + \frac{2}{x-2} \)[/tex]).
### Summary
- Addition, Subtraction, and Multiplication: These operations on polynomials always result in another polynomial. Hence, polynomials are closed under these operations.
- Division: This operation can yield rational functions, not necessarily polynomials. Therefore, polynomials are not closed under division.
I hope this detailed explanation helps in understanding the nature of polynomial operations and their closures!
