Which of the following responses shows that polynomials form a closed system

Question

under multiplication?(1 point)
$5\cdot \frac {3}{x}$
$(\frac {1}{3})(\frac {x}{4}-2)$
$2x^{3}+x^{2}$
$5x^{-1}\cdot 2$

Answer 1

To determine which of the responses shows that polynomials form a closed system under multiplication, we need to look for a product that results in another polynomial.

1. \(5\cdot \frac {3}{x}\)

   This expression simplifies to \(\frac{15}{x}\), which is not a polynomial (it has a negative exponent).

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

   This expression simplifies to \(\frac{1}{3} \cdot \frac{x}{4} - \frac{2}{3} = \frac{x}{12} - \frac{2}{3}\), which is a polynomial.

3. \(2x^{3}+x^{2}\)

   This expression is already a polynomial.

4. \(5x^{-1}\cdot 2\)

   This simplifies to \(\frac{10}{x}\), which is not a polynomial (again has a negative exponent).

Among the given options, \((\frac {1}{3})(\frac {x}{4}-2)\) and \(2x^{3}+x^{2}\) both yield a polynomial. However, to show that polynomials form a closed system under multiplication, we specifically look to ensure that the multiplication of polynomials results in another polynomial.

Thus, while \((\frac {1}{3})(\frac {x}{4}-2)\) is a polynomial, what we want to establish closure is policy agreement on both capital \(2x^{3}+x^{2}\), which already is a polynomial. Although the only option indicating that upon multiplication the final result remains a polynomial would point towards, \(2x^{3}+x^{2}\) specifically does not denote a product.

Conclusively, none of options represent a multiplication result clearly demonstrating polynomial closure. The only clear polynomial is option \(2x^{3}+x^{2}\).

In terms of closure under multiplication with polynomials, we should choose:

\(2x^{3}+x^{2}\)

to express that the expression is already polynomial confirming closure basis.

However since option defines what denotes closure indicate represents stating we observe closure under the multiplication strictly whenever \(a, b \in R[x]\) thus:

\(\boxed{2x^{3}+x^{2}}\) and proportional representation conveys polynomial closure under multiplication rightfully.

Answer 2

Thats wrong

Answer 3

I apologize for any confusion in my previous response. Let's clarify the situation regarding which of the provided options shows that polynomials form a closed system under multiplication.

To determine if a response shows that polynomials are closed under multiplication, we need to check if multiplying the given expressions results in another polynomial.

1. \(5\cdot \frac{3}{x}\):
   This simplifies to \(\frac{15}{x}\), which is not a polynomial because it has a negative exponent.

2. \((\frac{1}{3})(\frac{x}{4}-2)\):
   This simplifies to \(\frac{1}{3} \cdot \frac{x}{4} - \frac{1}{3} \cdot 2 = \frac{x}{12} - \frac{2}{3}\). This is a polynomial because it has non-negative integer exponents.

3. \(2x^{3}+x^{2}\):
   This is already a polynomial as it consists only of terms with non-negative integer exponents.

4. \(5x^{-1}\cdot 2\):
   This simplifies to \(\frac{10}{x}\), which is again not a polynomial due to the negative exponent.

To summarize:

• Option 1 and Option 4 do not result in polynomials after multiplication.
• Option 2 gives us a polynomial.
• Option 3 is a polynomial as stated.

However, in the context of showing closure under multiplication specifically, we look for a multiplication expression:

None of the options provided explicitly showcase a multiplication between two polynomials that yields another polynomial result. If we only consider explicitly multiplication of polynomials (for closure sake), only choices confirm a singular polynomial definition without direct multipliers.

The only option that depicts the definition of being polynomial without transformations or negative exponent results yet does not provide multiplication predicates is \(2x^{3} + x^{2}\) but doesn’t confirm multiplication closure.

Thus, the correct assessment for identifying a true multiplication product and closure under polynomial multiplication isn't displayed clearly in any of the expressions given as is, thus any choice lacks displaying a complete aspect of multiplication exemplification correctly.

If you seek a choice aiming at understanding polynomial definition while confirming closures would only reflect results framing foundational formulations, therefore:

If engaged on requirement, pick the one substantially having polynomial aspects:

Option: \(2x^{3} + x^{2}\) validates as polynomial no transformation
Acknowledge the polynomial closure however isn't directly shown through multiplication perspectives given in question hence valid options are also identified overall.