which of the following responses shows that polynomials form a closed system i=under multiplication

To determine whether polynomials form a closed system under multiplication, one must show that the product of any two polynomials is also a polynomial.

### Definition of a Closed System
A set is said to be closed under an operation (in this case, multiplication) if performing that operation on members of the set always produces a member of the same set.

### Polynomials
Polynomials are expressions composed of variables and coefficients combined using only addition, subtraction, multiplication, and non-negative integer exponents. A general polynomial in one variable \( x \) can be expressed as:

\[ P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \]

where \( a_n, a_{n-1}, ..., a_0 \) are coefficients, and \( n \) is a non-negative integer.

### Example of Polynomials under Multiplication
Let’s consider two polynomials:
1. \( P(x) = 2x^2 + 3x + 1 \)
2. \( Q(x) = x + 4 \)

### Multiplying \( P(x) \) and \( Q(x) \):
\[
P(x) \cdot Q(x) = (2x^2 + 3x + 1)(x + 4)
\]
Using distribution:
\[
= 2x^2 \cdot x + 2x^2 \cdot 4 + 3x \cdot x + 3x \cdot 4 + 1 \cdot x + 1 \cdot 4
\]
\[
= 2x^3 + 8x^2 + 3x^2 + 12x + x + 4
\]
Combining like terms:
\[
= 2x^3 + 11x^2 + 13x + 4
\]

### Result
The result \( 2x^3 + 11x^2 + 13x + 4 \) is again a polynomial.

### Conclusion
Since the product of any two polynomials is a polynomial, we can conclude that the set of polynomials is indeed closed under multiplication.

### Response
An appropriate response to illustrate this property might be:
"Polynomials form a closed system under multiplication because the product of any two polynomials is always a polynomial as demonstrated by multiplying \( P(x) = 2x^2 + 3x + 1 \) and \( Q(x) = x + 4 \), which yields the polynomial \( 2x^3 + 11x^2 + 13x + 4 \)."