To determine if the given polynomial forms a closed system under multiplication, we examine the expression provided:
\(-3(5x + 4y) = -15x - 12y\)
This expression shows a polynomial that results from multiplying a constant \(-3\) by the polynomial \(5x + 4y\).
Now let's analyze the options:
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Option #1: "Yes, because the result of multiplying the polynomials is also a polynomial."
- This statement is correct. The result \(-15x - 12y\) is indeed a polynomial, as it is a linear combination of the variables \(x\) and \(y\).
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Option #2: "No, because the exponent of x is not a positive integer."
- This statement is incorrect. While one term in a polynomial contains a variable raised to the first power, it is still a polynomial. A polynomial can include terms with non-negative integer exponents, and \(x^1\) is a valid term in a polynomial.
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Option #3: "No, because multiplying the polynomials resulted in subtraction."
- This statement is also incorrect. Subtraction is allowed in polynomial expressions, and forming a polynomial through operations that include subtraction does not preclude the result from being a polynomial.
Thus, the correct answer is Option #1: Yes, because the result of multiplying the polynomials is also a polynomial.