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
x squared plus 2 x plus 1
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
x squared plus 2 x plus 1
Answers
GPT-4o mini
To determine which of the responses demonstrates that polynomials form a closed system under multiplication, we need to verify whether the products of the pairs of expressions result in polynomials.
1. **(√x)(x+1)**:
- This expression results in \(x^{3/2} + \sqrt{x}\). The term \(x^{3/2}\) is a non-integer exponent, thus this is not a polynomial.
2. **(x²/2)(1/x)**:
- This expression simplifies to \(x^{2-1}/2 = x/2\). This is a polynomial because it can be expressed as \( \frac{1}{2}x \), which is a first-degree polynomial.
3. **(x² + 1)(x - 1/2)**:
- When multiplied, it results in \(x^3 - \frac{1}{2}x^2 + x - \frac{1}{2}\). This is also a polynomial (specifically, a cubic polynomial).
4. **x² + 2x + 1**:
- This is already in the standard form of a polynomial (specifically, a quadratic polynomial).
Considering the requirements to illustrate that polynomials form a closed system under multiplication, the second and third responses show multiplication of polynomials yielding polynomials. Therefore, the responses **(x²/2)(1/x)** and **(x² + 1)(x - 1/2)** demonstrate this closure property. Among these, the **third response** presents it most explicitly as a product of two polynomials.
Thus, the correct answer that most clearly demonstrates that polynomials form a closed system under multiplication is:
**(x² + 1)(x - 1/2)**.
1. **(√x)(x+1)**:
- This expression results in \(x^{3/2} + \sqrt{x}\). The term \(x^{3/2}\) is a non-integer exponent, thus this is not a polynomial.
2. **(x²/2)(1/x)**:
- This expression simplifies to \(x^{2-1}/2 = x/2\). This is a polynomial because it can be expressed as \( \frac{1}{2}x \), which is a first-degree polynomial.
3. **(x² + 1)(x - 1/2)**:
- When multiplied, it results in \(x^3 - \frac{1}{2}x^2 + x - \frac{1}{2}\). This is also a polynomial (specifically, a cubic polynomial).
4. **x² + 2x + 1**:
- This is already in the standard form of a polynomial (specifically, a quadratic polynomial).
Considering the requirements to illustrate that polynomials form a closed system under multiplication, the second and third responses show multiplication of polynomials yielding polynomials. Therefore, the responses **(x²/2)(1/x)** and **(x² + 1)(x - 1/2)** demonstrate this closure property. Among these, the **third response** presents it most explicitly as a product of two polynomials.
Thus, the correct answer that most clearly demonstrates that polynomials form a closed system under multiplication is:
**(x² + 1)(x - 1/2)**.