Which of the following shows that polynomials form a closed system under multiplication

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

Let’s denote two polynomials as follows:

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

where \(P(x)\) is a polynomial of degree \(n\) and \(Q(x)\) is a polynomial of degree \(m\), and \(a_i\) and \(b_j\) are coefficients (which can be real or complex numbers).

To find the product \(R(x) = P(x) \cdot Q(x)\), we calculate:

\[
R(x) = (a_n x^n + a_{n-1} x^{n-1} + \ldots + a_0)(b_m x^m + b_{m-1} x^{m-1} + \ldots + b_0)
\]

We distribute all terms of \(P(x)\) across all terms of \(Q(x)\). This results in:

\[
R(x) = \sum_{i=0}^{n} \sum_{j=0}^{m} a_i b_j x^{i+j}
\]

The degree of the resulting polynomial \(R(x)\) will be at most \(n + m\) (specifically, the leading term will be \(a_n \cdot b_m \cdot x^{n+m}\), assuming \(a_n \neq 0\) and \(b_m \neq 0\)).

The coefficients \(a_i b_j\) are also well-defined (since they are products of coefficients from the original polynomials), and they sum to form the resulting polynomial.

Thus, the product \(R(x)\) remains a polynomial, proving that the set of polynomials is closed under multiplication.

In conclusion, since the product of two polynomials yields another polynomial, we have shown that the polynomials form a closed system under multiplication.