To determine whether polynomials form a closed system under multiplication, we need to confirm that the product of any two polynomials is also a polynomial.
Let's examine the provided responses one by one:
- (1/2)x: This expression can be rewritten as \((1/2)x\), which is a polynomial of degree 1.
- (3x^2)/(4 - 1): Simplifying this gives \((3x^2)/(3) = x^2\), which is a polynomial of degree 2.
- 1/(4x^3) * 5: The term \(5/(4x^3)\) is not a polynomial, since it contains a term with a negative exponent when expressed as \(5 \cdot x^{-3}\).
- (sqrt(x))(6x * (-3)): The term \((\sqrt{x})\) can be expressed as \(x^{1/2}\). The expression then becomes \((6x(-3))(x^{1/2}) = -18x^{3/2}\), which again is not a polynomial because of the fractional exponent.
From this analysis, the responses clearly show that not all expressions are polynomials and specifically that (1/(4x^3) * 5) and (sqrt(x))(6x * (-3)) are not polynomials.
Therefore, polynomials do not form a closed system under multiplication because some products yield results that are not polynomials.