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)

Answer 1

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 \( (x)(1/2x) \).

Answer 2

you can only choose one

Answer 3

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).