Question
Which of the following responses shows that polynomials form a closed system under multiplication?(1 point)
Responses choose one that is the most correct
(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 choose one that is the most correct
(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 see if the product of the two expressions in each option is also a polynomial. A polynomial is an expression that consists of variables raised to non-negative integer powers and has coefficients that are real numbers.
Let's evaluate each response:
1. **(x)(12x)**: This is the product of two polynomials. The result is \(12x^2\), which is also a polynomial.
2. **(x)(Start Fraction 1 over 2 End Fraction x)**: This simplifies to \(\frac{1}{2} x^2\), which is again a polynomial.
3. **14x³⋅5**: This results in \(70x^3\), which is a polynomial.
4. **Start Fraction 1 over 4 x cubed End Fraction times 5**: This results in \(\frac{5}{4}x^3\), which is a polynomial.
5. **(x−−√)(6x ⋅−3)**: Here, \(\sqrt{x}\) is not a polynomial because it has a variable raised to a power that is not a non-negative integer. This product will not result in a polynomial.
6. **3x²⁴−1**: This expression appears to be a polynomial; however, it is ambiguous in structure. If it is interpreted as \(3x^{24}-1\), it is a polynomial.
Based on these evaluations, options 1, 2, 3, and 4 all yield results that are polynomials. However, the first response, **(x)(12x)**, is perhaps the simplest and clearly demonstrates closure under multiplication.
**Answer: (x)(12x)**.
Let's evaluate each response:
1. **(x)(12x)**: This is the product of two polynomials. The result is \(12x^2\), which is also a polynomial.
2. **(x)(Start Fraction 1 over 2 End Fraction x)**: This simplifies to \(\frac{1}{2} x^2\), which is again a polynomial.
3. **14x³⋅5**: This results in \(70x^3\), which is a polynomial.
4. **Start Fraction 1 over 4 x cubed End Fraction times 5**: This results in \(\frac{5}{4}x^3\), which is a polynomial.
5. **(x−−√)(6x ⋅−3)**: Here, \(\sqrt{x}\) is not a polynomial because it has a variable raised to a power that is not a non-negative integer. This product will not result in a polynomial.
6. **3x²⁴−1**: This expression appears to be a polynomial; however, it is ambiguous in structure. If it is interpreted as \(3x^{24}-1\), it is a polynomial.
Based on these evaluations, options 1, 2, 3, and 4 all yield results that are polynomials. However, the first response, **(x)(12x)**, is perhaps the simplest and clearly demonstrates closure under multiplication.
**Answer: (x)(12x)**.