Let's carefully examine each of the given equations to determine whether the set of polynomials is closed under a specific operation.
Option A: Addition
[tex]\[
(x^2 + x) + (x + 1) = x^2 + 2x + 1
\][/tex]
Adding two polynomials results in another polynomial. Therefore, this set of polynomials is closed under addition.
Option B: Subtraction
[tex]\[
(3x^4 + x^3) - (-2x^4 + x^3) = 3x^4 + x^3 + 2x^4 - x^3 = 5x^4
\][/tex]
Subtracting one polynomial from another results in another polynomial. Hence, the set of polynomials is closed under subtraction.
Option C: Division
[tex]\[
\frac{x^2 - 5x + 3}{x - 2} = x - 3 + \frac{-3}{x - 2}
\][/tex]
In this case, dividing one polynomial by another does not necessarily result in a polynomial because the remainder term, [tex]\(\frac{-3}{x - 2}\)[/tex], introduces a rational function. Thus, the set of polynomials is not closed under division.
Option D: Multiplication
[tex]\[
(x^2 - 5x + 3)(x - 5) = x^3 - 5x^2 - 5x^2 + 25x + 3x - 15 = x^3 - 10x^2 + 28x - 15
\][/tex]
Multiplying two polynomials results in another polynomial. Thus, the set of polynomials is closed under multiplication.
The correct answer is:
C. Division: [tex]\(\frac{x^2 - 5x + 3}{x - 2} = x - 3 + \frac{-3}{x - 2}\)[/tex]
This demonstrates that the set of polynomials is not closed under division because the result is not necessarily a polynomial.
