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

Question

Which of the following responses demonstrates that polynomials form a closed system under multiplication?
(1 point)
• x2 + 2x + 1
O (VT) (x + 1)
• (22 + 1) (I - })
• (2) (4)

GPT-4o mini · Answer

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

Let's analyze the options provided:

$x^{2} + 2 x + 1$ : This is a polynomial by itself, but it does not demonstrate a multiplication of two polynomials.
$(x + 1)$ : This is also a polynomial but does not show a product.
$(2 x + 1) (x - 1)$ : This is a product of two polynomials. When multiplied, it yields $2 x^{2} - 2 x + x - 1 = 2 x^{2} - x - 1$ , which is also a polynomial.
$(2) (4)$ : This involves multiplication of two constants, which are technically polynomials of degree 0. However, it does not demonstrate the multiplication of two non-constant polynomials.

The response that best demonstrates the closure of polynomials under multiplication is:

(2x + 1)(x - 1).