To determine whether the given expression forms a closed system under multiplication, we first need to analyze the components provided.
The expression: \[ -3(5x + 4y) = -\frac{15}{x} - 12y \] suggests that the left side of the equation is a polynomial while the right side contains a term \(-\frac{15}{x}\), which is not a polynomial, since polynomials cannot have negative exponents (a term like \(x^{-1}\) does not satisfy the criteria for being a polynomial).
Let's evaluate the options:
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Option #1: Yes, because the result of multiplying the polynomials is also a polynomial.
This option is incorrect; the right-hand side includes a term that is not a polynomial. -
Option #2: No, because the exponent of x is not a positive integer.
This option is correct; the term \(-\frac{15}{x}\) implies \(x^{-1}\), which does not meet the polynomial definition. -
Option #3: No, because multiplying the polynomials resulted in subtraction.
This option is misleading; the operation does not affect whether it's a polynomial.
Based on the evaluation, the correct answer is Option #2. The system is not closed under multiplication because the result involves a term (\(-\frac{15}{x}\)) that is not a polynomial. Therefore, the final answer is:
Option #2: No, because the exponent of x is not a positive integer.
1 answer