Question

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

Answer 1

To determine if polynomials form a closed system under multiplication, we need to consider whether the product of any two polynomials results in another polynomial.

Here are the general characteristics of the responses that would indicate that polynomials are closed under multiplication:

1. **Definition of Closure**: A set is said to be closed under a certain operation if performing that operation on elements within the set always produces an element that is also part of the set.

2. **Example with Polynomials**: If you multiply two polynomials, say \( P(x) = x^2 + 2 \) and \( Q(x) = x + 3 \), their product \( P(x) \cdot Q(x) = (x^2 + 2)(x + 3) = x^3 + 3x^2 + 2x + 6 \) is still a polynomial.

3. **General Claim**: You could provide a statement like "For any two polynomials \( P(x) \) and \( Q(x) \), their product \( R(x) = P(x) \cdot Q(x) \) is also a polynomial," which illustrates that the multiplication of two polynomials results in another polynomial.

4. **Formal Proof or Statement**: A formal statement like "The product of any two polynomials is a polynomial, hence the set of polynomials is closed under multiplication" would solidify this concept.

If any of these elements are included in the responses you are considering, it would indicate that polynomials do indeed form a closed system under multiplication.