Question

Which of the following options correctly describes if the following polynomial forms a closed system under multiplication? - 3(5/x + 4y) = - 15/x - 12y Option #1: Yes, because the result of multiplying the polynomials is also a polynomial. Option #2: No, because the exponent of x is not a positive integer. Option #3: No, because multiplying the polynomials resulted in subtraction. (1 point) Option #

Answer 1

To determine whether the expression given forms a closed system under multiplication and which option correctly describes it, we need to consider the properties of polynomial forms.

Given the expression:

\[
-3\left(\frac{5}{x} + 4y\right) = -\frac{15}{x} - 12y
\]

This expression includes a term with \(\frac{1}{x}\), which is not a polynomial term since polynomials only allow for non-negative integer exponents for their variables. Thus, the term \(\frac{1}{x}\) can be rewritten as \(x^{-1}\), and since the exponent is -1 (which is not a positive integer), this indicates that we do not have a polynomial.

Now, we can analyze the options:

- **Option #1: Yes, because the result of multiplying the polynomials is also a polynomial.**
This option is incorrect because the expression is not a polynomial due to the term involving \( \frac{1}{x} \).

- **Option #2: No, because the exponent of \(x\) is not a positive integer.**
This option is correct, as the term \(\frac{1}{x}\) has a negative exponent.

- **Option #3: No, because multiplying the polynomials resulted in subtraction.**
This option is incorrect because subtraction does not affect whether it's a polynomial; rather, it depends on the form of the expression.

Given this analysis, the correct option is:

**Option #2: No, because the exponent of \(x\) is not a positive integer.**