Question

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

(x)(12x)
left parenthesis x right parenthesis left parenthesis Start Fraction 1 over 2 End Fraction x right parenthesis

14x3⋅5
Start Fraction 1 over 4 x cubed End Fraction times 5

3x24−1
Start Fraction 3 x squared over 4 minus 1 End Fraction

(x−−√)(6x ⋅−3 )

Answer 1

To determine which response shows that polynomials form a closed system under multiplication, we need to identify which of the provided options results in a polynomial when multiplied.

1. \( (x)(12x) \) = \( 12x^2 \) (a polynomial)
2. \( \left(x\right) \left(\frac{1}{2} x\right) \) = \( \frac{1}{2} x^2 \) (a polynomial)
3. \( \frac{3 x^2}{4} - 1 \) represents a polynomial, but it's not a multiplication operation.
4. \( (x - \sqrt{})(6x \cdot -3) \) = ?

The product needs to be analyzed:
- \( x - \sqrt{} \) is not a polynomial due to the square root.
- Therefore, the multiplication of \( (x - \sqrt{}) \) with any polynomial will not yield a polynomial.

From the evaluation above, the responses that result in polynomials are:

- \( (x)(12x) \)
- \( \left(x\right) \left(\frac{1}{2} x\right) \)

Thus, both these options show that polynomials are closed under multiplication since the result of the products are polynomials.

To summarize:
The responses that demonstrate the closure of polynomials under multiplication are:

1. \( (x)(12x) \)
2. \( \left(x\right) \left(\frac{1}{2} x\right) \)

However, if you need only one response, either of the first two will suffice as showing closure.