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+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
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
x2+2x+1
Answers
GPT-4o mini
To demonstrate that polynomials form a closed system under multiplication, we need to show that when multiplying two polynomials, the result is also a polynomial.
Let's analyze the options provided:
1. **\((\sqrt{x})(x + 1)\)**: The term \(\sqrt{x}\) is not a polynomial because polynomials can only have non-negative integer exponents. Thus, this product does not demonstrate closure.
2. **\((x^2 + 1)(x - \frac{1}{2})\)**: Both \(x^2 + 1\) and \(x - \frac{1}{2}\) are polynomials. When we multiply them, the result will also be a polynomial. Therefore, this example demonstrates closure under multiplication.
3. **\(x^2 + 2x + 1\)**: This expression is a polynomial, but it does not illustrate the multiplication of two polynomials to show closure; rather, it's already a single polynomial expression.
The best response that shows polynomials form a closed system under multiplication is:
**\((x^2 + 1)(x - \frac{1}{2})\)** - both are polynomials, and their product is also a polynomial, confirming closure.
Let's analyze the options provided:
1. **\((\sqrt{x})(x + 1)\)**: The term \(\sqrt{x}\) is not a polynomial because polynomials can only have non-negative integer exponents. Thus, this product does not demonstrate closure.
2. **\((x^2 + 1)(x - \frac{1}{2})\)**: Both \(x^2 + 1\) and \(x - \frac{1}{2}\) are polynomials. When we multiply them, the result will also be a polynomial. Therefore, this example demonstrates closure under multiplication.
3. **\(x^2 + 2x + 1\)**: This expression is a polynomial, but it does not illustrate the multiplication of two polynomials to show closure; rather, it's already a single polynomial expression.
The best response that shows polynomials form a closed system under multiplication is:
**\((x^2 + 1)(x - \frac{1}{2})\)** - both are polynomials, and their product is also a polynomial, confirming closure.