Question

Which of the following responses shows that polynomials form a closed system
under multiplication?(1 point)
$\frac {3x^{2}}{4-1}$
$(x)(\frac {1}{2}x)$
$(\sqrt {x})(6x\cdot -3)$
. $\frac {1}{4x^{3}}\cdot 5$

Answer 1

To determine if polynomials form a closed system under multiplication, we need to analyze the responses given to verify whether they result in polynomials.

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

Simplifying this expression, we have:
\[
\frac{3x^{2}}{4-1} = \frac{3x^{2}}{3} = x^{2}
\]
This is a polynomial.

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

Multiplying these terms, we get:
\[
(x)(\frac{1}{2}x) = \frac{1}{2}x^2
\]
This is also a polynomial.

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

First, simplify \(6x \cdot -3\):
\[
6x \cdot -3 = -18x
\]
Now multiplying \(\sqrt{x}\) and \(-18x\):
\[
(\sqrt{x})(-18x) = -18\sqrt{x} \cdot x = -18x^{3/2}
\]
This is not a polynomial since \(\sqrt{x} = x^{1/2}\) has a fractional exponent.

4. **Response: \(\frac{1}{4x^{3}} \cdot 5\)**

Simplifying this gives:
\[
\frac{5}{4x^{3}} = \frac{5}{4}x^{-3}
\]
This is not a polynomial because it has a negative exponent.

From the above analysis, we see that only the first two responses yield polynomials. However, since the third and fourth responses result in non-polynomial expressions, we conclude that polynomials do not form a closed system under multiplication.

The responses show that:
- The first two are polynomials,
- The third and fourth are not.

Thus, the conclusion is that polynomials are not closed under multiplication based on the given options.

Answer 2

Pick one of the answers

Answer 3

The response that shows polynomials form a closed system under multiplication is:

**\((x)(\frac{1}{2}x)\)**

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