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

3x24−1
Start Fraction 3 x squared over 4 minus 1 End Fraction

(x−−√)(6x ⋅−3 )
open paren square root of x close paren times open paren 6 x times negative 3 close paren

14x3⋅5
Start Fraction 1 over 4 x cubed End Fraction times 5

Answer 1

Pick one

Answer 2

To determine which response shows that polynomials form a closed system under multiplication, we look for expressions that result in a polynomial after multiplication.

1. **(x)(12x)**: This simplifies to \(12x^2\), which is a polynomial.

2. **(x)(\frac{1}{2}x)**: This simplifies to \(\frac{1}{2}x^2\), which is also a polynomial.

3. **\(\frac{3x^2}{4} - 1\)**: This is not a multiplication response; it's a polynomial, but it doesn't demonstrate closure under multiplication.

4. **(\(\sqrt{x}\))(6x \cdot -3)**: The first term, \(\sqrt{x}\), is not a polynomial (as polynomials can only have non-negative integer exponents). Therefore, the product of these terms is not a polynomial.

5. **\(\frac{1}{4}x^3 \cdot 5\)**: This simplifies to \(\frac{5}{4}x^3\), which is a polynomial.

The responses that show closure under multiplication are:
- **(x)(12x) → 12x²**
- **(x)(\frac{1}{2}x) → \frac{1}{2}x²**
- **\(\frac{1}{4}x^3 \cdot 5→ \frac{5}{4}x^3\)**

The first two responses show polynomials multiplying to result in another polynomial. The clearest response indicating closure under multiplication would be either of the first two.

Answer 3

I would choose **(x)(12x)**, which simplifies to \(12x^2\). This response clearly shows that multiplying two polynomials (in this case, \(x\) and \(12x\)) results in another polynomial, demonstrating that polynomials form a closed system under multiplication.