To demonstrate that polynomials form a closed system under multiplication, we need to consider the resulting expressions when we multiply two polynomials together.
Let's evaluate each option:
A. \((x^2/2)(1/x)\)
- Here, \((x^2/2)\) is not a polynomial since it involves a constant multiplied by \(x^2\), but \(1/x\) is not a polynomial either. Therefore, this is not a valid polynomial multiplication and does not demonstrate closure.
B. \(x^2 + 2x + 1\)
- This is not a multiplication of two polynomials; it is a single polynomial expression. Therefore, it does not directly demonstrate the closure property.
C. \((x^2 + 1)(x - 1/2)\)
- Here, both \((x^2 + 1)\) and \((x - 1/2)\) are polynomials. When we multiply these two polynomials, the result will indeed be a polynomial. For example, the multiplication would yield:
\[ (x^2 + 1)(x - 1/2) = x^2 \cdot x + 1 \cdot x - (1/2)x^2 - (1/2) = x^3 - (1/2)x^2 + x - (1/2) \]
The result is still a polynomial.
Thus, the choice that demonstrates that polynomials form a closed system under multiplication is:
C. (x^2 + 1)(x – 1/2)
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