Question
Which of the following responses demonstrates that polynomials form a closed system under multiplication?(1 point) Responses x^2+2x+1 x squared plus 2 x plus 1 \left(\frac{x^2}{2}\right)\left(\frac{1}{x}\right) left parenthesis Start Fraction x squared over 2 End Fraction right parenthesis left parenthesis Start Fraction 1 over x End Fraction right parenthesis \left(x^2+1\right)\left(x-\frac{1}{2}\right) left parenthesis x squared plus 1 right parenthesis left parenthesis x minus Start Fraction 1 over 2 End Fraction right parenthesis \left(\sqrt{x}\right)\left(x+1\right)
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GPT-4o mini
To determine which response demonstrates that polynomials form a closed system under multiplication, we need to multiply the expressions provided and check if the result is a polynomial.
1. **Response: \( x^2 + 2x + 1 \)**
- This is already a polynomial.
2. **Response: \( \left(\frac{x^2}{2}\right)\left(\frac{1}{x}\right) \)**
- When we multiply:
\[
\frac{x^2}{2} \cdot \frac{1}{x} = \frac{x^{2}}{2x} = \frac{x}{2}
\]
- This is a polynomial (specifically a linear polynomial).
3. **Response: \( \left(x^2 + 1\right)\left(x - \frac{1}{2}\right) \)**
- When we expand:
\[
(x^2 + 1)(x - \frac{1}{2}) = x^3 - \frac{1}{2}x^2 + x - \frac{1}{2}
\]
- This is a polynomial (specifically a cubic polynomial).
4. **Response: \( \left(\sqrt{x}\right)\left(x + 1\right) \)**
- When we multiply:
\[
\sqrt{x} \cdot (x + 1) = x^{\frac{3}{2}} + \sqrt{x}
\]
- This result is not a polynomial, as polynomials do not allow for fractional exponents.
**Conclusion:** The responses that demonstrate that polynomials form a closed system under multiplication are:
- \( x^2 + 2x + 1 \)
- \( \left(\frac{x^2}{2}\right)\left(\frac{1}{x}\right) \)
- \( \left(x^2 + 1\right)\left(x - \frac{1}{2}\right) \)
However, the first two responses specifically show that polynomials can be multiplied to produce a valid polynomial product, confirming closure. The best example to select from the provided options is the one involving \( \left(x^2 + 1\right)\left(x - \frac{1}{2}\right) \) because it clearly involves the multiplication of two polynomials resulting in another polynomial.
1. **Response: \( x^2 + 2x + 1 \)**
- This is already a polynomial.
2. **Response: \( \left(\frac{x^2}{2}\right)\left(\frac{1}{x}\right) \)**
- When we multiply:
\[
\frac{x^2}{2} \cdot \frac{1}{x} = \frac{x^{2}}{2x} = \frac{x}{2}
\]
- This is a polynomial (specifically a linear polynomial).
3. **Response: \( \left(x^2 + 1\right)\left(x - \frac{1}{2}\right) \)**
- When we expand:
\[
(x^2 + 1)(x - \frac{1}{2}) = x^3 - \frac{1}{2}x^2 + x - \frac{1}{2}
\]
- This is a polynomial (specifically a cubic polynomial).
4. **Response: \( \left(\sqrt{x}\right)\left(x + 1\right) \)**
- When we multiply:
\[
\sqrt{x} \cdot (x + 1) = x^{\frac{3}{2}} + \sqrt{x}
\]
- This result is not a polynomial, as polynomials do not allow for fractional exponents.
**Conclusion:** The responses that demonstrate that polynomials form a closed system under multiplication are:
- \( x^2 + 2x + 1 \)
- \( \left(\frac{x^2}{2}\right)\left(\frac{1}{x}\right) \)
- \( \left(x^2 + 1\right)\left(x - \frac{1}{2}\right) \)
However, the first two responses specifically show that polynomials can be multiplied to produce a valid polynomial product, confirming closure. The best example to select from the provided options is the one involving \( \left(x^2 + 1\right)\left(x - \frac{1}{2}\right) \) because it clearly involves the multiplication of two polynomials resulting in another polynomial.
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