Question
Which of the following responses demonstrates that polynomials form a closed system under multiplication?(1 point) Responses x2+2x+1 x squared plus 2 x plus 1 (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 (x−−√)(x+1) left parenthesis Start Root x End Root right parenthesis left parenthesis x plus 1 right parenthesis (x2+1)(x−12)
Answers
Answered by
GPT-4o mini
To demonstrate that polynomials form a closed system under multiplication, we need to show that the product of any two polynomial expressions is also a polynomial.
Let's evaluate the provided responses one by one:
1. **\(x^2 + 2x + 1\)**: This is a polynomial.
2. **\((x^2)(1x)\)**: This simplifies to \(x^3\), which is also a polynomial.
3. **\(\left( \frac{x^2}{2} \right) \left( \frac{1}{x} \right)\)**: This simplifies to \(\frac{x^2}{2x} = \frac{x}{2}\), which is a polynomial (though not in standard form). It’s important to note that this expression includes division by \(x\), which can lead to problems at \(x=0\), so it's technically not a polynomial since polynomials cannot have variables in the denominator.
4. **\((\sqrt{x})(x + 1)\)**: This can be rewritten as \(x^{1/2}(x + 1)\). The term \(\sqrt{x}\) indicates that there is a fractional exponent, which means this expression is not a polynomial.
5. **\((x^2 + 1)(x - 12)\)**: This is a multiplication of two polynomials. The result will be a polynomial of degree 3, confirming the closure under multiplication.
From this evaluation, the responses that illustrate the closure property for polynomials under multiplication are:
- \(x^2 + 2x + 1\)
- \((x^2)(1x)\)
- \((x^2 + 1)(x - 12)\)
The response that validates the closure property under multiplication unambiguously is **\((x^2 + 1)(x - 12)\)**, as both factors are polynomials, and their product is guaranteed to be a polynomial.
Thus, the **best response** demonstrating that polynomials form a closed system under multiplication is \((x^2 + 1)(x - 12)\).
Let's evaluate the provided responses one by one:
1. **\(x^2 + 2x + 1\)**: This is a polynomial.
2. **\((x^2)(1x)\)**: This simplifies to \(x^3\), which is also a polynomial.
3. **\(\left( \frac{x^2}{2} \right) \left( \frac{1}{x} \right)\)**: This simplifies to \(\frac{x^2}{2x} = \frac{x}{2}\), which is a polynomial (though not in standard form). It’s important to note that this expression includes division by \(x\), which can lead to problems at \(x=0\), so it's technically not a polynomial since polynomials cannot have variables in the denominator.
4. **\((\sqrt{x})(x + 1)\)**: This can be rewritten as \(x^{1/2}(x + 1)\). The term \(\sqrt{x}\) indicates that there is a fractional exponent, which means this expression is not a polynomial.
5. **\((x^2 + 1)(x - 12)\)**: This is a multiplication of two polynomials. The result will be a polynomial of degree 3, confirming the closure under multiplication.
From this evaluation, the responses that illustrate the closure property for polynomials under multiplication are:
- \(x^2 + 2x + 1\)
- \((x^2)(1x)\)
- \((x^2 + 1)(x - 12)\)
The response that validates the closure property under multiplication unambiguously is **\((x^2 + 1)(x - 12)\)**, as both factors are polynomials, and their product is guaranteed to be a polynomial.
Thus, the **best response** demonstrating that polynomials form a closed system under multiplication is \((x^2 + 1)(x - 12)\).
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