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

Question

Responses

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

14x3⋅5
1
4
𝑥
3
⋅
5
Start Fraction 1 over 4 x cubed End Fraction times 5

3x24−1
3
𝑥
2
4
−
1
Start Fraction 3 x squared over 4 minus 1 End Fraction

(x−−√)(6x ⋅−3 )

Answer 1

To determine which of the given responses shows that polynomials form a closed system under multiplication, we need to identify which result of the multiplication is still a polynomial.

Polynomials are defined as expressions that involve non-negative integer powers of the variable (e.g., \(x^n\), where \(n\) is a non-negative integer).

Now let's evaluate each option:

1. \((x)(12x)\):

   • This results in \(12x^2\), which is a polynomial.

2. \((x)(\frac{1}{2}x)\):

   • This results in \(\frac{1}{2}x^2\), which is also a polynomial.

3. \(\frac{3x^2}{4} - 1\):

   • This expression is actually a polynomial, but it is not a product. It consists of a polynomial term and a constant term.

4. \((\sqrt{x})(6x \cdot -3)\):

   • This is \(\sqrt{x} \cdot (6x \cdot -3) = -18x \sqrt{x}\). Here, \(\sqrt{x}\) is not a polynomial term since it has a fractional exponent (\(x^{1/2}\)). Therefore, this is not a polynomial.

The first two options are valid polynomial multiplications, illustrating that the multiplication of polynomials results in another polynomial. Therefore, both \((x)(12x)\) and \((x)(\frac{1}{2}x)\) demonstrate that polynomials are closed under multiplication.

So, the best responses showing closure under multiplication are (x)(12x) or (x)(\frac{1}{2}x).