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Which of the following responses shows that polynomials form a closed system under multiplication?(1 point) Responses (x)(12x)...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
14x3⋅5
Start Fraction 1 over 4 x cubed End Fraction times 5
(x−−√)(6x ⋅−3 )
open paren square root of x close paren times open paren 6 x times negative 3 close paren
3x24−1
Responses
(x)(12x)
left parenthesis x right parenthesis left parenthesis Start Fraction 1 over 2 End Fraction x right parenthesis
14x3⋅5
Start Fraction 1 over 4 x cubed End Fraction times 5
(x−−√)(6x ⋅−3 )
open paren square root of x close paren times open paren 6 x times negative 3 close paren
3x24−1
Answers
GPT-4o mini
To determine which response shows that polynomials form a closed system under multiplication, we need to identify responses that result in a polynomial after performing the multiplication.
1. **(x)(12x)**: This results in \(12x^2\), which is a polynomial.
2. **\( (x)(\frac{1}{2}x) \)**: This results in \(\frac{1}{2}x^2\), which is a polynomial.
3. **\( \frac{1}{4}x^3 \cdot 5 \)**: This results in \(\frac{5}{4}x^3\), which is a polynomial.
4. **\( (\sqrt{x})(6x \cdot -3) \)**: This results in \(-18x^{\frac{3}{2}}\), which is not a polynomial because it has a fractional exponent.
5. **\( 3x^{24} - 1 \)**: This represents a polynomial expression, but it doesn't show multiplication since it's just a polynomial term.
Based on these analyses, the responses that demonstrate closure under multiplication (resulting in a polynomial) are:
1. **(x)(12x)**
2. **\( (x)(\frac{1}{2}x) \)**
3. **\( \frac{1}{4}x^3 \cdot 5 \)**
Therefore, the correct answers that show that polynomials form a closed system under multiplication are the first three responses.
1. **(x)(12x)**: This results in \(12x^2\), which is a polynomial.
2. **\( (x)(\frac{1}{2}x) \)**: This results in \(\frac{1}{2}x^2\), which is a polynomial.
3. **\( \frac{1}{4}x^3 \cdot 5 \)**: This results in \(\frac{5}{4}x^3\), which is a polynomial.
4. **\( (\sqrt{x})(6x \cdot -3) \)**: This results in \(-18x^{\frac{3}{2}}\), which is not a polynomial because it has a fractional exponent.
5. **\( 3x^{24} - 1 \)**: This represents a polynomial expression, but it doesn't show multiplication since it's just a polynomial term.
Based on these analyses, the responses that demonstrate closure under multiplication (resulting in a polynomial) are:
1. **(x)(12x)**
2. **\( (x)(\frac{1}{2}x) \)**
3. **\( \frac{1}{4}x^3 \cdot 5 \)**
Therefore, the correct answers that show that polynomials form a closed system under multiplication are the first three responses.