Question

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

(x−−√)(6x ⋅−3 )
open paren square root of x close paren times open paren 6 x times negative 3 close paren

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

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

14x3⋅5

Answer 1

To determine which response shows that polynomials form a closed system under multiplication, we need to look for an expression where multiplying polynomials results in another polynomial.

Let's examine each response:

1. \((\sqrt{x})(6x \cdot -3)\): This involves a square root, which is not a polynomial. Thus, this does not show closure.

2. \(\frac{3x^2}{4} - 1\): This expression is itself a polynomial, but it doesn't represent multiplication of polynomials.

3. \((x)(12x)\): This is the multiplication of two polynomials (since \(x\) is a polynomial and \(12x\) is also a polynomial). The product, \(12x^2\), is also a polynomial. This demonstrates that polynomials are closed under multiplication.

4. \(14x^3 \cdot 5\): This is the multiplication of a polynomial \(14x^3\) and a constant \(5\), resulting in \(70x^3\), which is also a polynomial. This also demonstrates closure.

Both options 3 and 4 show closure under multiplication.

**Final answer:**
- \((x)(12x)\) (Option 3) demonstrates that polynomials form a closed system under multiplication.