To determine if polynomials form a closed system under multiplication, we need to see if the product of two polynomial expressions results in another polynomial.
Let's analyze each of the given options:
A) \( \frac{3x^2}{4} - 1 \): This is a polynomial itself, but it is not a product of two polynomials.
B) \( (x)(\frac{1}{2}x) \): The product of two polynomials \( x \) and \( \frac{1}{2}x \) is \( \frac{1}{2}x^2 \), which is a polynomial.
C) \( (\sqrt{x})(6x^{-3}) \): Here, \( \sqrt{x} \) is \( x^{1/2} \) and \( 6x^{-3} \) is \( 6/x^3 \). The product \( 6x^{-2.5} \) (which is not a polynomial since it has a negative exponent) indicates that polynomials are not closed under this multiplication.
D) \( \frac{1}{4}x^3 \times 5 \): This results in \( \frac{5}{4}x^3 \), which is a polynomial.
From these assessments, the response that demonstrates polynomials are closed under multiplication is:
B) \( (x)(\frac{1}{2}x) \)
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