Question
Which of the following responses demonstrates that polynomials form a closed system under multiplication?(1 point)
Responses
(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)
left parenthesis x squared plus 1 right parenthesis left parenthesis x minus Start Fraction 1 over 2 End Fraction right parenthesis
x2+2x+1
ABC OR D
Responses
(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)
left parenthesis x squared plus 1 right parenthesis left parenthesis x minus Start Fraction 1 over 2 End Fraction right parenthesis
x2+2x+1
ABC OR D
Answers
Answer
IS IT 123 OR THE 4TH
Answered by
GPT-4o mini
To determine which response demonstrates that polynomials form a closed system under multiplication, we need to check whether the product of the given expressions results in a polynomial.
Let's analyze each response:
A) \((x^2)(1x)\): The product is \(x^3\), which is a polynomial.
B) \((\sqrt{x})(x+1)\): The product is \(x^{3/2} + \sqrt{x}\), which is not a polynomial because it contains a fractional exponent.
C) \((x^2 + 1)(x - \frac{1}{2})\): The product results in a polynomial, specifically \(x^3 - \frac{1}{2}x^2 + x - \frac{1}{2}\).
D) \(x^2 + 2x + 1\): This is already a polynomial.
Therefore, the responses A, C, and D provide products that are polynomials. However, the most direct response that demonstrates closure under multiplication specifically is A \((x^2)(1x)\) or C \((x^2 + 1)(x - \frac{1}{2})\).
Given your options and the question, I would conclude that:
**Response A** is sufficient to show that polynomials form a closed system under multiplication.
So the answer is **A**.
Let's analyze each response:
A) \((x^2)(1x)\): The product is \(x^3\), which is a polynomial.
B) \((\sqrt{x})(x+1)\): The product is \(x^{3/2} + \sqrt{x}\), which is not a polynomial because it contains a fractional exponent.
C) \((x^2 + 1)(x - \frac{1}{2})\): The product results in a polynomial, specifically \(x^3 - \frac{1}{2}x^2 + x - \frac{1}{2}\).
D) \(x^2 + 2x + 1\): This is already a polynomial.
Therefore, the responses A, C, and D provide products that are polynomials. However, the most direct response that demonstrates closure under multiplication specifically is A \((x^2)(1x)\) or C \((x^2 + 1)(x - \frac{1}{2})\).
Given your options and the question, I would conclude that:
**Response A** is sufficient to show that polynomials form a closed system under multiplication.
So the answer is **A**.
Answered by
GPT-4o mini
To determine if the options 1, 2, or 3 demonstrate closure of polynomials under multiplication, or if the 4th option does, let's clarify the options based on your previous message:
1. **(x²)(1x)**: This is a multiplication of two polynomials, resulting in \(x^3\), which is a polynomial.
2. **(√x)(x + 1)**: This involves \(√x\), which is not a polynomial (because it can be represented as \(x^{1/2}\), a non-integer exponent). The result \(x^{3/2} + \sqrt{x}\) is also not a polynomial.
3. **(x² + 1)(x - ½)**: This is a multiplication of two polynomials, resulting in a polynomial.
4. **x² + 2x + 1**: This is already a polynomial.
Given the criterion that a polynomial must have all its terms with non-negative integer exponents, we find that:
- Option 1 (A) \( \text{(x²)(1x)} \) = \(x^3\) (a polynomial)
- Option 2 (B) \( \text{(√x)(x + 1)} \) = not a polynomial
- Option 3 (C) \( \text{(x² + 1)(x - ½)} \) = a polynomial
- Option 4 (D) \( \text{x² + 2x + 1} \) = a polynomial
Thus, only options 1, 3, and 4 demonstrate closure under multiplication (they yield polynomials).
Since you asked if it’s options 1, 2, or 3, or the 4th option, the correct answer is that options **1, 3, and 4** demonstrate closure under multiplication.
If you're asking for a singular choice that most clearly demonstrates closure under multiplication from the original statement, **Option 1 or Option 3** are the best representatives since they specifically show multiplication of two polynomials resulting in a polynomial. The 4th option doesn't demonstrate multiplication, it just states a polynomial.
Thus, you can confidently choose **1, 3, and 4** but if you have to select just one to illustrate the concept of closure through multiplication, pick **1 or 3**.
1. **(x²)(1x)**: This is a multiplication of two polynomials, resulting in \(x^3\), which is a polynomial.
2. **(√x)(x + 1)**: This involves \(√x\), which is not a polynomial (because it can be represented as \(x^{1/2}\), a non-integer exponent). The result \(x^{3/2} + \sqrt{x}\) is also not a polynomial.
3. **(x² + 1)(x - ½)**: This is a multiplication of two polynomials, resulting in a polynomial.
4. **x² + 2x + 1**: This is already a polynomial.
Given the criterion that a polynomial must have all its terms with non-negative integer exponents, we find that:
- Option 1 (A) \( \text{(x²)(1x)} \) = \(x^3\) (a polynomial)
- Option 2 (B) \( \text{(√x)(x + 1)} \) = not a polynomial
- Option 3 (C) \( \text{(x² + 1)(x - ½)} \) = a polynomial
- Option 4 (D) \( \text{x² + 2x + 1} \) = a polynomial
Thus, only options 1, 3, and 4 demonstrate closure under multiplication (they yield polynomials).
Since you asked if it’s options 1, 2, or 3, or the 4th option, the correct answer is that options **1, 3, and 4** demonstrate closure under multiplication.
If you're asking for a singular choice that most clearly demonstrates closure under multiplication from the original statement, **Option 1 or Option 3** are the best representatives since they specifically show multiplication of two polynomials resulting in a polynomial. The 4th option doesn't demonstrate multiplication, it just states a polynomial.
Thus, you can confidently choose **1, 3, and 4** but if you have to select just one to illustrate the concept of closure through multiplication, pick **1 or 3**.
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