Question
Which of the following responses shows that polynomials form a closed system under multiplication?(1 point) Responses (x−−√)(6x ⋅−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 14x3⋅5 Start Fraction 1 over 4 x cubed End Fraction times 5 (x)(1/2x)
Answers
GPT-4o mini
To determine which of the given responses shows that polynomials form a closed system under multiplication, we need to check if the result of multiplying two polynomials is also a polynomial.
1. **(√x)(6x ⋅ -3)**:
- Here, √x is not a polynomial (it's a radical), so the product does not guarantee that it will give a polynomial.
2. **(3x²/4 − 1)**:
- This expression is a polynomial in itself. However, it's not showing a multiplication of two separate polynomials.
3. **(14x³)(5) or (1/4)x³ * 5**:
- 14x³ and 5 are both polynomials. Their product (14x³ * 5 = 70x³) is also a polynomial.
- Similarly, (1/4)x³ * 5 results in (5/4)x³, which is also a polynomial.
4. **(x)(1/2x)**:
- Here, both x and 1/2x are polynomials. Their multiplication results in (1/2)x², which is also a polynomial.
From these responses, the following show that polynomials form a closed system under multiplication:
- **(14x³)(5)** or **(1/4)x³ * 5**
- **(x)(1/2x)**
The best responses indicating closure under multiplication would be \( (14x^3)(5) \) or \( (x)(1/2x) \).
1. **(√x)(6x ⋅ -3)**:
- Here, √x is not a polynomial (it's a radical), so the product does not guarantee that it will give a polynomial.
2. **(3x²/4 − 1)**:
- This expression is a polynomial in itself. However, it's not showing a multiplication of two separate polynomials.
3. **(14x³)(5) or (1/4)x³ * 5**:
- 14x³ and 5 are both polynomials. Their product (14x³ * 5 = 70x³) is also a polynomial.
- Similarly, (1/4)x³ * 5 results in (5/4)x³, which is also a polynomial.
4. **(x)(1/2x)**:
- Here, both x and 1/2x are polynomials. Their multiplication results in (1/2)x², which is also a polynomial.
From these responses, the following show that polynomials form a closed system under multiplication:
- **(14x³)(5)** or **(1/4)x³ * 5**
- **(x)(1/2x)**
The best responses indicating closure under multiplication would be \( (14x^3)(5) \) or \( (x)(1/2x) \).
Honeybeee
you can only choose one
GPT-4o mini
If I can only choose one response that clearly demonstrates that polynomials form a closed system under multiplication, I would select:
**(14x³)(5)**
This response involves the multiplication of two polynomials (14x³ and 5), and the result is also a polynomial (70x³).
**(14x³)(5)**
This response involves the multiplication of two polynomials (14x³ and 5), and the result is also a polynomial (70x³).