To determine which of the responses shows that polynomials form a closed system under multiplication, we need to clarify what it means for polynomials to be closed under multiplication. A set is considered closed under an operation if performing that operation on any elements of the set produces an element that also belongs to the set.
Polynomials are expressions composed of variables raised to non-negative integer powers and multiplied by coefficients. Therefore, if the product of two polynomials yields another polynomial, this demonstrates closure under multiplication.
Let's analyze each option:
A. \((x)(\frac{1}{2}x)\)
- This is a product of two polynomials. It simplifies to \(\frac{1}{2}x^2\), which is a polynomial. Hence, this shows closure under multiplication.
B. \(\frac{1}{4x^3} \cdot 5\)
- This expression includes \(\frac{1}{x^3}\), which is not a polynomial because polynomials cannot have negative powers of the variable (i.e., terms like \(x^{-3}\)). Therefore, this option does not show closure.
C. \(\frac{3x^2}{4-1}\)
- This simplifies to \(\frac{3x^2}{3}\) or \(x^2\), which is a polynomial. However, the format of the expression suggests that the division by a constant may mislead one to consider closure, but \( \frac{3x^2}{3} \) does yield a polynomial. Thus, it can be considered a polynomial, but the question phrasing is ambiguous.
D. \((\sqrt{x})(6x \cdot -3)\)
- The term \(\sqrt{x}\) can be expressed as \(x^{1/2}\), which is not a polynomial because polynomials only include non-negative integer powers. Therefore, this option does not show closure.
Thus, the best options that clearly show closure under multiplication of polynomials are:
A. \((x)(\frac{1}{2}x)\)
and potentially
C. \(\frac{3x^2}{4-1}\) as it results in a polynomial.
However, A is the clearest example of closure under multiplication of polynomials.
1 answer