To show that polynomials form a closed system under multiplication, we need to demonstrate that the product of any two polynomial expressions is also a polynomial.
Letβs evaluate each of the listed responses:
1. **\(14x^3 \cdot 5\)**
- This is the multiplication of a polynomial \(14x^3\) (with degree 3) and a constant \(5\). The result will be \(70x^3\), which is a polynomial.
2. **\(\frac{3x^2}{4} - 1\)**
- This expression is a polynomial (specifically, a rational polynomial). However, it does not demonstrate multiplication.
3. **\((x)(\frac{1}{2} x)\)**
- This is the multiplication of two polynomials: \(x\) and \(\frac{1}{2} x\), which yields \(\frac{1}{2} x^2\), a polynomial.
4. **\((\sqrt{x})(6x \cdot -3)\)**
- The term \(\sqrt{x}\) is not a polynomial, since polynomials only include non-negative integer exponents. Thus, this expression does not demonstrate closure.
The first three responses contain polynomial expressions, and their products result in a polynomial. However, the best response demonstrating that polynomials form a closed system under multiplication is:
**\(14x^3 \cdot 5\)**, because it clearly shows two polynomial terms being multiplied and producing another polynomial result.
Answer:
**14xΒ³ β
5** (which yields \(70x^3\), a polynomial).