To determine which response shows that polynomials form a closed system under multiplication, we must check if the result of multiplying the terms is still a polynomial.
Let's analyze each option:
A. \( \frac{3x^2}{4} - 1 \) - This is a polynomial, but it is not a product of two polynomials. It is simply an expression.
B. \( (\sqrt{x})(6x \cdot -3) \) - Here, \( \sqrt{x} \) is not a polynomial, as polynomials only involve non-negative integer powers of \( x \). So, this doesn't show closure under multiplication.
C. \( (x)(\frac{1}{2}x) \) - Both terms are polynomials, and their product is \( \frac{1}{2}x^2 \), which is also a polynomial. This is a valid demonstration of closure under multiplication.
D. \( \frac{1}{4}x^3 \cdot 5 \) - Here, both terms are indeed polynomials (a constant is considered a polynomial of degree 0), and their product is \( \frac{5}{4}x^3 \), which is also a polynomial. This is also a valid demonstration of closure under multiplication.
Thus, responses C and D show properties that confirm closure under multiplication in polynomials.
So, the most straightforward answer that directly shows the closure with respect to the multiplication of polynomials is:
C. (x)(1/2x)
1 answer