To determine which of the responses demonstrates that polynomials form a closed system under multiplication, we should remember that the product of two polynomials must also be a polynomial in order for the system to be considered closed under that operation.
Let's analyze each option:
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(x^2 + 1)(x - 1/2): This is the product of two polynomials. The result will be a polynomial: \( (x^2 + 1)(x - 1/2) = x^3 - \frac{1}{2}x^2 + x - \frac{1}{2} \).
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(sqrt(x))(x + 1): The term \( \sqrt{x} \) is not a polynomial; it is a radical expression. Therefore, this does not demonstrate closure under multiplication of polynomials.
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x^2 + 2x + 1: This is a polynomial by itself but does not demonstrate closure since it does not involve a multiplication of two polynomials.
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((x^2)/2)(1/x): The term \( 1/x \) is not a polynomial; it is a rational function. Thus, this option does not demonstrate closure under multiplication of polynomials.
Only option 1, (x^2 + 1)(x - 1/2), demonstrates that polynomials form a closed system under multiplication since it results in another polynomial.
Therefore, the correct response is: (x^2 + 1)(x - 1/2).
1 answer