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 x2+2x+1 x squared plus 2 x plus 1 (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)
Answers
GPT-4o mini
To determine if the provided responses demonstrate that polynomials form a closed system under multiplication, we need to check if the product of two polynomials results in another polynomial.
Let's examine the responses:
1. \((\sqrt{x})(x + 1)\)
- This expression includes \(\sqrt{x}\), which is not a polynomial since it contains a non-integer exponent (1/2).
2. \(x^2 + 2x + 1\)
- This is a polynomial (specifically, a quadratic polynomial).
3. \((x^2 + 1)(x - \frac{1}{2})\)
- This expression is the product of two polynomials, and the result will also be a polynomial. Thus, it demonstrates the closure of polynomials under multiplication.
4. \((x^2)(1/x)\)
- This expression simplifies to \(x\), which is a polynomial, but since \(1/x\) is not a polynomial, the original product does not demonstrate closure.
Based on this analysis, the response that clearly demonstrates that polynomials form a closed system under multiplication is:
**\((x^2 + 1)(x - \frac{1}{2})\)**.
Let's examine the responses:
1. \((\sqrt{x})(x + 1)\)
- This expression includes \(\sqrt{x}\), which is not a polynomial since it contains a non-integer exponent (1/2).
2. \(x^2 + 2x + 1\)
- This is a polynomial (specifically, a quadratic polynomial).
3. \((x^2 + 1)(x - \frac{1}{2})\)
- This expression is the product of two polynomials, and the result will also be a polynomial. Thus, it demonstrates the closure of polynomials under multiplication.
4. \((x^2)(1/x)\)
- This expression simplifies to \(x\), which is a polynomial, but since \(1/x\) is not a polynomial, the original product does not demonstrate closure.
Based on this analysis, the response that clearly demonstrates that polynomials form a closed system under multiplication is:
**\((x^2 + 1)(x - \frac{1}{2})\)**.