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Which of the following responses demonstrates that polynomials form a closed system under multiplication?(1 point) Responses (x...Asked by grann
Which of the following responses demonstrates that polynomials form a closed system under multiplication?(1 point) Responses (x−−√)(x+1) left parenthesis Start Root x End Root right parenthesis left parenthesis x plus 1 right parenthesis (x2+1)(x−12) left parenthesis x squared plus 1 right parenthesis left parenthesis x minus Start Fraction 1 over 2 End Fraction right parenthesis (x22)(1x) left parenthesis Start Fraction x squared over 2 End Fraction right parenthesis left parenthesis Start Fraction 1 over x End Fraction right parenthesis x2+2x+1
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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:
1. \((x^{1/2})(x + 1)\) - This contains \(x^{1/2}\), which is not a polynomial (it's a root).
2. \((x^2 + 1)(x - 12)\) - Both \(x^2 + 1\) and \(x - 12\) are polynomials. Their product will also be a polynomial.
3. \((\frac{x^2}{2})(\frac{1}{x})\) - Here, while the terms involved are derived from polynomial expressions, the product simplifies to \(\frac{x}{2}\), which is a polynomial.
4. \(x^2 + 2x + 1\) - 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.
Among the given options:
1. \((x^{1/2})(x + 1)\) - This contains \(x^{1/2}\), which is not a polynomial (it's a root).
2. \((x^2 + 1)(x - 12)\) - Both \(x^2 + 1\) and \(x - 12\) are polynomials. Their product will also be a polynomial.
3. \((\frac{x^2}{2})(\frac{1}{x})\) - Here, while the terms involved are derived from polynomial expressions, the product simplifies to \(\frac{x}{2}\), which is a polynomial.
4. \(x^2 + 2x + 1\) - 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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