Question
Which of the following responses shows that polynomials form a closed system under multiplication?(1 point) Responses (x)(12x) left parenthesis x right parenthesis left parenthesis Start Fraction 1 over 2 End Fraction x right parenthesis (x−−√)(6x ⋅−3 ) open paren square root of x close paren times open paren 6 x times negative 3 close paren 3x24−1 Start Fraction 3 x squared over 4 minus 1 End Fraction 14x3⋅5
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Answer
Please choose one correct answer
Answered by
GPT-4o mini
To determine which of the given responses shows that polynomials form a closed system under multiplication, we need to find the response where the multiplication of two polynomial expressions results in another polynomial.
Let's evaluate the responses one by one:
1. **(x)(12x)**:
- This equals \( 12x^2 \), which is a polynomial.
2. **(x)(\frac{1}{2}x)**:
- This equals \( \frac{1}{2}x^2 \), which is also a polynomial.
3. **(\sqrt{x})(6x \cdot -3)**:
- This equals \( \sqrt{x} \cdot -18x \) which is \( -18x\sqrt{x} \). Since \(\sqrt{x}\) is not a polynomial (it is a radical), this does not represent a polynomial.
4. **(3x^2 / 4 - 1)**:
- This expression is already a polynomial (it is a polynomial of degree 2).
5. **(14x^3 \cdot 5)**:
- This equals \( 70x^3 \), which is also a polynomial.
From the evaluations above, responses (1), (2), (4), and (5) demonstrate the closure of polynomials under multiplication.
However, if the question requires a single response that shows this property, responses (1) and (2) explicitly illustrate polynomial multiplication yielding polynomials. Therefore, both (1) and (2) are valid representations of closure under multiplication of polynomials.
If you are looking for just one response that exemplifies closure, either **(x)(12x)** or **(x)(\frac{1}{2}x)** can be chosen.
Let's evaluate the responses one by one:
1. **(x)(12x)**:
- This equals \( 12x^2 \), which is a polynomial.
2. **(x)(\frac{1}{2}x)**:
- This equals \( \frac{1}{2}x^2 \), which is also a polynomial.
3. **(\sqrt{x})(6x \cdot -3)**:
- This equals \( \sqrt{x} \cdot -18x \) which is \( -18x\sqrt{x} \). Since \(\sqrt{x}\) is not a polynomial (it is a radical), this does not represent a polynomial.
4. **(3x^2 / 4 - 1)**:
- This expression is already a polynomial (it is a polynomial of degree 2).
5. **(14x^3 \cdot 5)**:
- This equals \( 70x^3 \), which is also a polynomial.
From the evaluations above, responses (1), (2), (4), and (5) demonstrate the closure of polynomials under multiplication.
However, if the question requires a single response that shows this property, responses (1) and (2) explicitly illustrate polynomial multiplication yielding polynomials. Therefore, both (1) and (2) are valid representations of closure under multiplication of polynomials.
If you are looking for just one response that exemplifies closure, either **(x)(12x)** or **(x)(\frac{1}{2}x)** can be chosen.
Answered by
GPT-4o mini
The correct response that shows that polynomials form a closed system under multiplication is:
**(x)(12x)**
This results in \( 12x^2 \), which is a polynomial.
**(x)(12x)**
This results in \( 12x^2 \), which is a polynomial.
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