Question
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 (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+1)(x−12) left parenthesis x squared plus 1 right parenthesis left parenthesis x minus Start Fraction 1 over 2 End Fraction right parenthesis x2+2x+1
Answers
Answered by
GPT-4o mini
To demonstrate that polynomials form a closed system under multiplication, we need to show that when you multiply two polynomials, the result is also a polynomial.
Let’s examine the offered responses one by one:
1. **\((\sqrt{x})(x+1)\)**:
Here, \(\sqrt{x}\) is not a polynomial since it is a radical expression. Therefore, this product is not guaranteed to be a polynomial.
2. **\((x^2)(\frac{1}{x})\)**:
This simplifies to \(x^{2-1} = x\), which is indeed a polynomial. However, it involves a division by \(x\), which could lead to non-polynomial terms in general situations.
3. **\((x^2 + 1)(x - \frac{1}{2})\)**:
Both \(x^2 + 1\) and \(x - \frac{1}{2}\) are polynomials. The multiplication of two polynomials results in another polynomial. This demonstrates the closure of polynomials under multiplication.
4. **\(x^2 + 2x + 1\)**:
This is already a polynomial on its own.
Given the responses, the one that clearly demonstrates closure under multiplication is **\((x^2 + 1)(x - \frac{1}{2})\)**, as it explicitly shows two polynomials being multiplied, resulting in another polynomial.
Thus, the correct response is: **\((x^2 + 1)(x - \frac{1}{2})\)**.
Let’s examine the offered responses one by one:
1. **\((\sqrt{x})(x+1)\)**:
Here, \(\sqrt{x}\) is not a polynomial since it is a radical expression. Therefore, this product is not guaranteed to be a polynomial.
2. **\((x^2)(\frac{1}{x})\)**:
This simplifies to \(x^{2-1} = x\), which is indeed a polynomial. However, it involves a division by \(x\), which could lead to non-polynomial terms in general situations.
3. **\((x^2 + 1)(x - \frac{1}{2})\)**:
Both \(x^2 + 1\) and \(x - \frac{1}{2}\) are polynomials. The multiplication of two polynomials results in another polynomial. This demonstrates the closure of polynomials under multiplication.
4. **\(x^2 + 2x + 1\)**:
This is already a polynomial on its own.
Given the responses, the one that clearly demonstrates closure under multiplication is **\((x^2 + 1)(x - \frac{1}{2})\)**, as it explicitly shows two polynomials being multiplied, resulting in another polynomial.
Thus, the correct response is: **\((x^2 + 1)(x - \frac{1}{2})\)**.
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