Which of the following responses demonstrates the polynomials form a closed system under the multiplication

To demonstrate that polynomials form a closed system under multiplication, we need to show that if you take any two polynomials and multiply them together, the result is also a polynomial.

Let’s define two general polynomials:

1. \( P(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \)
2. \( Q(x) = b_m x^m + b_{m-1} x^{m-1} + \ldots + b_1 x + b_0 \)

where \( a_n, a_{n-1}, \ldots, a_0 \) and \( b_m, b_{m-1}, \ldots, b_0 \) are coefficients, and \( n \) and \( m \) are the degrees of the respective polynomials.

Now, when we multiply these two polynomials, we get:

\[
P(x) \cdot Q(x) = (a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0)(b_m x^m + b_{m-1} x^{m-1} + \ldots + b_1 x + b_0)
\]

When expanding this product using the distributive property, each term in \( P(x) \) will be multiplied by each term in \( Q(x) \). The highest degree of the resultant polynomial will be \( n + m \) (the sum of the degrees of the two polynomials).

Since multiplying two terms of the form \( a_i x^i \) and \( b_j x^j \) (where \( a_i \) and \( b_j \) are coefficients) results in a term of the form \( (a_i b_j) x^{i+j} \), which is still in the form of a polynomial, we conclude that the product of any two polynomials is also a polynomial.

Thus, we can state that the set of all polynomials is closed under multiplication.

If you have specific options that demonstrate this property, please provide them, and I can help assess which response fits best.