Question
Which of the following responses shows that polynomials form a closed system under multiplication?(1 point)
Responses
3xexponent2/4−1
Start Fraction 3 x squared over 4 minus 1 End Fraction
(x−−√)(6x ⋅−3 )
open paren square root of x close paren times open paren 6 x times negative 3 close paren
(x)(1/2 x)
left parenthesis x right parenthesis left parenthesis Start Fraction 1 over 2 End Fraction x right parenthesis
1/4xexponent3 ⋅5
Start Fraction 1 over 4 x cubed End Fraction times 5
is it 123 or 4?
Responses
3xexponent2/4−1
Start Fraction 3 x squared over 4 minus 1 End Fraction
(x−−√)(6x ⋅−3 )
open paren square root of x close paren times open paren 6 x times negative 3 close paren
(x)(1/2 x)
left parenthesis x right parenthesis left parenthesis Start Fraction 1 over 2 End Fraction x right parenthesis
1/4xexponent3 ⋅5
Start Fraction 1 over 4 x cubed End Fraction times 5
is it 123 or 4?
Answers
Answer
did you make shure you counted the "closed system'' part?
Answered by
GPT-4o mini
To determine which response shows that polynomials form a closed system under multiplication, we need to check if the result of the multiplication in each option is also a polynomial.
1. **Response 1:** \(\frac{3x^2}{4} - 1\)
This expression is a polynomial.
2. **Response 2:** \((\sqrt{x})(6x \cdot -3)\)
This simplifies to \(-18x^{3/2}\), which is not a polynomial because polynomials can't have fractional (non-integer) exponents.
3. **Response 3:** \((x)(\frac{1}{2} x)\)
This simplifies to \(\frac{1}{2}x^2\), which is a polynomial.
4. **Response 4:** \(\frac{1}{4}x^3 \cdot 5\)
This simplifies to \(\frac{5}{4}x^3\), which is also a polynomial.
Responses 1, 3, and 4 result in polynomials, but Response 2 does not.
Since we are looking for responses that demonstrate closure under multiplication, Responses 3 and 4 clearly show that multiplying two polynomials (in this case, \(x\) and \(\frac{1}{2}x\) as well as \(\frac{1}{4}x^3\) and \(5\)) results in another polynomial.
Hence, the correct responses showing closure under multiplication would be Response 3 and Response 4. If you're prompted to pick only one, it appears that Response 4 (1/4x^3 ⋅ 5) provides a direct multiplication of two elements resulting in a polynomial form that clearly represents closure.
So, if you are to choose one, go with **Response 4**.
1. **Response 1:** \(\frac{3x^2}{4} - 1\)
This expression is a polynomial.
2. **Response 2:** \((\sqrt{x})(6x \cdot -3)\)
This simplifies to \(-18x^{3/2}\), which is not a polynomial because polynomials can't have fractional (non-integer) exponents.
3. **Response 3:** \((x)(\frac{1}{2} x)\)
This simplifies to \(\frac{1}{2}x^2\), which is a polynomial.
4. **Response 4:** \(\frac{1}{4}x^3 \cdot 5\)
This simplifies to \(\frac{5}{4}x^3\), which is also a polynomial.
Responses 1, 3, and 4 result in polynomials, but Response 2 does not.
Since we are looking for responses that demonstrate closure under multiplication, Responses 3 and 4 clearly show that multiplying two polynomials (in this case, \(x\) and \(\frac{1}{2}x\) as well as \(\frac{1}{4}x^3\) and \(5\)) results in another polynomial.
Hence, the correct responses showing closure under multiplication would be Response 3 and Response 4. If you're prompted to pick only one, it appears that Response 4 (1/4x^3 ⋅ 5) provides a direct multiplication of two elements resulting in a polynomial form that clearly represents closure.
So, if you are to choose one, go with **Response 4**.
Answered by
GPT-4o mini
Thank you for pointing that out. Let's clarify what it means for a set to be a "closed system" under an operation such as multiplication.
A set is closed under an operation (like multiplication) if, when you take any two elements from that set and apply the operation, the result is also an element of the same set.
In the context of polynomials and the responses provided:
1. **Response 1:** \(\frac{3x^2}{4} - 1\)
- This is a polynomial, but it doesn’t demonstrate multiplication.
2. **Response 2:** \((\sqrt{x})(6x \cdot -3)\)
- Results in \(-18x^{3/2}\), which is not a polynomial. Thus, it does not support closure in the context of polynomials.
3. **Response 3:** \((x)(\frac{1}{2}x)\)
- Results in \(\frac{1}{2}x^2\), which is a polynomial. This illustrates closure since multiplying polynomials results in another polynomial.
4. **Response 4:** \(\frac{1}{4}x^3 \cdot 5\)
- Results in \(\frac{5}{4}x^3\), which is also a polynomial. This again exemplifies closure.
To definitively demonstrate that the set of polynomials is closed under multiplication, we need responses that show two polynomials being multiplied together to produce another polynomial.
Given this reasoning, **Responses 3 and 4** do illustrate closure under multiplication since the result of multiplying polynomials yields polynomials.
If you specifically need to choose only one response to exemplify closure under multiplication, **Response 4** is a clear example as it directly illustrates the multiplication of a polynomial by a constant, producing another polynomial.
Thank you for your patience, and I hope this clarifies the aspect of closure!
A set is closed under an operation (like multiplication) if, when you take any two elements from that set and apply the operation, the result is also an element of the same set.
In the context of polynomials and the responses provided:
1. **Response 1:** \(\frac{3x^2}{4} - 1\)
- This is a polynomial, but it doesn’t demonstrate multiplication.
2. **Response 2:** \((\sqrt{x})(6x \cdot -3)\)
- Results in \(-18x^{3/2}\), which is not a polynomial. Thus, it does not support closure in the context of polynomials.
3. **Response 3:** \((x)(\frac{1}{2}x)\)
- Results in \(\frac{1}{2}x^2\), which is a polynomial. This illustrates closure since multiplying polynomials results in another polynomial.
4. **Response 4:** \(\frac{1}{4}x^3 \cdot 5\)
- Results in \(\frac{5}{4}x^3\), which is also a polynomial. This again exemplifies closure.
To definitively demonstrate that the set of polynomials is closed under multiplication, we need responses that show two polynomials being multiplied together to produce another polynomial.
Given this reasoning, **Responses 3 and 4** do illustrate closure under multiplication since the result of multiplying polynomials yields polynomials.
If you specifically need to choose only one response to exemplify closure under multiplication, **Response 4** is a clear example as it directly illustrates the multiplication of a polynomial by a constant, producing another polynomial.
Thank you for your patience, and I hope this clarifies the aspect of closure!
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