Question
Which of the following responses shows that polynomials form a closed system under multiplication?(1 point)
Responses
(x−−√)(6x x−3 )
open paren square root of x close paren times open paren 6 x times negative 3 close paren
3x^2/4−1
Start Fraction 3 x squared over 4 minus 1 End Fraction
1/4x^3⋅ 5
Start Fraction 1 over 4 x cubed End Fraction times 5
(x)(1/2x)
Responses
(x−−√)(6x x−3 )
open paren square root of x close paren times open paren 6 x times negative 3 close paren
3x^2/4−1
Start Fraction 3 x squared over 4 minus 1 End Fraction
1/4x^3⋅ 5
Start Fraction 1 over 4 x cubed End Fraction times 5
(x)(1/2x)
Answers
GPT-4o mini
To demonstrate that polynomials form a closed system under multiplication, we need to ensure that multiplying two polynomials results in another polynomial. Let's analyze each of the provided expressions:
1. **\((\sqrt{x})(6x(-3))\)**:
- This involves \(\sqrt{x}\), which is not a polynomial because it contains a fractional exponent (1/2). Therefore, this expression does not show closure under multiplication for polynomials.
2. **\(\frac{3x^2}{4} - 1\)**:
- This expression is a polynomial (specifically, it represents a polynomial of degree 2). However, it does not demonstrate multiplication but rather subtraction.
3. **\(\frac{1}{4}x^3 \cdot 5\)**:
- This represents the multiplication of a polynomial term \(\frac{1}{4}x^3\) by a constant \(5\), which results in a polynomial \(\frac{5}{4}x^3\). This shows closure under multiplication.
4. **\((x)(\frac{1}{2}x)\)**:
- This multiplication results in \(\frac{1}{2}x^2\), which is also a polynomial. This demonstrates closure under multiplication as well.
Considering the responses for closure under multiplication of polynomials, both option 3 and option 4 show that the product remains a polynomial. If you are to select only one response that shows polynomials form a closed system under multiplication, either option 3 or option 4 would be correct.
For the sake of clarity, the best response demonstrating closure under multiplication is:
\((x)(\frac{1}{2}x)\)
1. **\((\sqrt{x})(6x(-3))\)**:
- This involves \(\sqrt{x}\), which is not a polynomial because it contains a fractional exponent (1/2). Therefore, this expression does not show closure under multiplication for polynomials.
2. **\(\frac{3x^2}{4} - 1\)**:
- This expression is a polynomial (specifically, it represents a polynomial of degree 2). However, it does not demonstrate multiplication but rather subtraction.
3. **\(\frac{1}{4}x^3 \cdot 5\)**:
- This represents the multiplication of a polynomial term \(\frac{1}{4}x^3\) by a constant \(5\), which results in a polynomial \(\frac{5}{4}x^3\). This shows closure under multiplication.
4. **\((x)(\frac{1}{2}x)\)**:
- This multiplication results in \(\frac{1}{2}x^2\), which is also a polynomial. This demonstrates closure under multiplication as well.
Considering the responses for closure under multiplication of polynomials, both option 3 and option 4 show that the product remains a polynomial. If you are to select only one response that shows polynomials form a closed system under multiplication, either option 3 or option 4 would be correct.
For the sake of clarity, the best response demonstrating closure under multiplication is:
\((x)(\frac{1}{2}x)\)