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 3x24−1 Start Fraction 3 x squared over 4 minus 1 End Fraction (x−−√)(6x ⋅−3 )

Answer 1

To demonstrate that polynomials form a closed system under multiplication, we need to show that multiplying two polynomials together results in another polynomial.

Let's examine each response:

1. **(x)(12x)**: This is a multiplication of two polynomials (x and 12x), resulting in \(12x^2\), which is a polynomial. This response shows closure.

2. **(x)(\(\frac{1}{2}x\))**: This also results in \(\frac{1}{2}x^2\), which is a polynomial. This response shows closure.

3. **\(14x^3 \cdot 5 \cdot \frac{1}{4}x^3\)**: This simplifies to \( \frac{14 \cdot 5}{4} x^6\) or \(\frac{70}{4} x^6\), which is \(17.5 x^6\), a polynomial. This response shows closure.

4. **\(3x^2 \cdot 4 - 1\)**: This represents a polynomial \(12x^2 - 1\), which is a polynomial. This response shows closure as well.

5. **\((\sqrt{x})(6x \cdot -3)\)**: The term \(\sqrt{x}\) is not a polynomial; therefore, this combination does not show closure, as \(\sqrt{x}\) is not a polynomial.

The responses that show polynomials form a closed system under multiplication are:
1. (x)(12x)
2. (x)(\(\frac{1}{2}x\))
3. \(14x^3 \cdot 5 \cdot \frac{1}{4}x^3\)
4. \(3x^2 \cdot 4 - 1\)

The response containing \((\sqrt{x})(6x \cdot -3)\) does not demonstrate closure because \(\sqrt{x}\) is not a polynomial.