Question

Which of the following responses shows that polynomials form a closed system under multiplication?(1 point)
Responses

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 )
open paren square root of x close paren times open paren 6 x times negative 3 close paren

(x)(12x)

Answer 1

To demonstrate that polynomials form a closed system under multiplication, we need to show that when we multiply any two polynomials, the result is still a polynomial.

Let's analyze the provided responses:

1. **\(14x^3 \cdot 5\)**: This is a multiplication of a polynomial \(14x^3\) (which is a polynomial of degree 3) by a constant \(5\) (also a polynomial of degree 0). The result \(70x^3\) is a polynomial, so this shows closure under multiplication.

2. **\(\frac{3x^2}{4} - 1\)**: This is not a multiplication but rather a polynomial expression. It does not demonstrate the closure property.

3. **\((\sqrt{x})(6x \cdot -3)\)**: Although \(6x \cdot -3\) is polynomial (specifically, \(-18x\)), \(\sqrt{x}\) is not a polynomial, as polynomials cannot have fractional exponents. Hence, this does not demonstrate closure.

4. **\((x)(12x)\)**: This is the multiplication of \(x\) (a polynomial of degree 1) by \(12x\) (also a polynomial of degree 1). The result is \(12x^2\), which is also a polynomial. This shows closure under multiplication.

The best responses showing that polynomials form a closed system under multiplication are:

- \(14x^3 \cdot 5\)
- \((x)(12x)\)

Both of these responses demonstrate closure since the result is still a polynomial after multiplication.