To demonstrate that polynomials form a closed system under multiplication, we need to show that when we multiply any two polynomials, the result is also a polynomial.
Among the given options:
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\((x^{1/2})(x + 1)\) - This contains \(x^{1/2}\), which is not a polynomial (it's a root).
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\((x^2 + 1)(x - 12)\) - Both \(x^2 + 1\) and \(x - 12\) are polynomials. Their product will also be a polynomial.
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\((\frac{x^2}{2})(\frac{1}{x})\) - Here, while the terms involved are derived from polynomial expressions, the product simplifies to \(\frac{x}{2}\), which is a polynomial.
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\(x^2 + 2x + 1\) - This is a polynomial by itself, but it doesn't demonstrate closure with another polynomial since it is a standalone expression.
The response that demonstrates closure under multiplication is:
\((x^2 + 1)(x - 12)\)
This indicates that the product of the two polynomials is also a polynomial, thereby establishing that polynomials form a closed system under multiplication.
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