To determine which of the given options demonstrates that the set of polynomials is not closed under a certain operation, we need to understand what closure means in the context of polynomials. A set is closed under an operation if applying that operation to any two elements of the set results in another element that is also within that set.
Polynomials are expressions involving variables and coefficients, where the variables are raised to non-negative integer powers, and the set of polynomials includes all such expressions.
Let's examine each option:
A. Addition:
[tex]\[
(3x^4 + x^3) + (-2x^4 + x^3) = x^4 + 2x^3
\][/tex]
Adding two polynomials results in another polynomial. Hence, polynomials are closed under addition.
B. Division:
[tex]\[
(x^2 + 2x) / (x + 1)
\][/tex]
Dividing one polynomial by another does not necessarily result in a polynomial. For example, [tex]\( (x^2 + 2x)/(x + 1) \)[/tex] can be simplified to [tex]\( x + 1 + \frac{1}{x+1} \)[/tex], which includes a term of [tex]\( \frac{1}{x+1} \)[/tex]. This term is not a polynomial since polynomials cannot have variables in the denominator.
C. Multiplication:
[tex]\[
(x^2 + 2x)(x + 1) = x^3 + 3x^2 + 2x
\][/tex]
Multiplying two polynomials results in another polynomial. Therefore, polynomials are closed under multiplication.
D. Multiplication:
[tex]\[
(3x^4 + x^3)(-2x^4 + x^3) = -6x^8 + x^7 + x^6
\][/tex]
Similar to option C, multiplying two polynomials results in another polynomial. Hence, polynomials are closed under multiplication.
From the above analysis, we see that division is the operation under which the set of polynomials is not closed.
Therefore, the correct answer is:
B. Division
\textrightarrow This demonstrates that the set of polynomials is not closed under the operation of division.
