Question

Which of the following responses shows that polynomials form a closed system under multiplication?(1 point)
Responses

5x−1⋅2
5x−1⋅2
- incorrect

2x3+x2
2 x cubed plus x squared - no response given

(13)(x4−2)
1 third times open paren x over 4 minus 2 close paren - no response given

5⋅3x

Answer 1

To determine which response shows that polynomials form a closed system under multiplication, we need to recognize that multiplying two polynomials always produces another polynomial.

Let’s analyze the provided options:

1. **5x−1⋅2**: This represents the multiplication of a polynomial \(5x - 1\) and a constant (which can also be considered a polynomial of degree 0). The result will be another polynomial. Hence, this shows closure.

2. **2x³ + x²**: This is already a polynomial and doesn't represent multiplication.

3. **(1/3)(x⁴ - 2)**: Here, \(1/3\) is a constant, and multiplying it by the polynomial \(x^4 - 2\) results in another polynomial. Thus, this shows closure as well.

4. **5⋅3x**: This also represents multiplication of constants, resulting in a constant polynomial. Therefore, it also shows closure.

In summary, options **5x−1⋅2**, **(1/3)(x⁴ - 2)**, and **5⋅3x** all demonstrate that polynomials form a closed system under multiplication. However, only the first and third explicitly show a polynomial multiplied by another term, while the second is simply a polynomial, and the fourth is a product of constants.

Thus, the responses **5x−1⋅2** and **(1/3)(x⁴ - 2)** are proper representations of closure under multiplication for polynomials. If you want to select one, the response of **5x−1⋅2** directly illustrates this with two polynomial-like components.