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:

$$-3(\frac{5}{x}+4y)=-\frac{15}{x}-12y$$

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.