To determine which of the responses demonstrates that polynomials form a closed system under multiplication, we need to see if the results of the multiplications presented yield a polynomial.
A polynomial is an expression that can be written in the form \( a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \) where \( a_i \) are constants (coefficients) and \( n \) is a non-negative integer.
Let's analyze each option:
A) \( (x^2 + 1)(x - \frac{1}{2}) \)
This is the product of two polynomials. The result will be: \[ x^2 \cdot x - \frac{1}{2} x^2 + 1 \cdot x - \frac{1}{2} = x^3 - \frac{1}{2}x^2 + x - \frac{1}{2} \] This expression is a polynomial.
B) \( (x)(x + 1) \)
This is also the product of two polynomials. The result will be: \[ x^2 + x \] This expression is a polynomial.
C) \( x^2 + 2x + 1 \)
This expression is already given as a polynomial and does not involve a multiplication operation.
D) \( \left(\frac{x^2}{2}\right) \left(\frac{1}{x}\right) \)
This is the product of two expressions, but \( \frac{1}{x} \) is not a polynomial. The result would be: \[ \frac{x^2}{2} \cdot \frac{1}{x} = \frac{x}{2} \] While \( \frac{x}{2} \) is a polynomial, if we consider that \( \frac{1}{x} \) is not a polynomial, this does not demonstrate closure under multiplication for polynomials.
From the options provided, both A and B demonstrate that polynomials form a closed system under multiplication, as their results are polynomials. However, since the question likely expects only one response:
Final Answer: A) \((x^2 + 1)(x - \frac{1}{2})\) or B) \((x)(x + 1)\) (both demonstrate closure). If forced to select one: A.
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