Question

Which of the following responses demonstrates that polynomials form a closed system under multiplication?

A, (x^2+1)(x-1/2)

B, (√x)(x+1)

C, (x^2/2)(1/x)

D, x^2+2x+1

Answer 1

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

Let's evaluate each option:

A. \((x^2+1)(x-1/2)\)

- Both \(x^2 + 1\) and \(x - 1/2\) are polynomials. Multiplying them will yield another polynomial.
- Result: This represents a polynomial.

B. \((\sqrt{x})(x+1)\)

- Here, \(\sqrt{x}\) can be rewritten as \(x^{1/2}\), which is not a polynomial since polynomials must have whole number (non-negative integer) exponents.
- Result: This does not represent a polynomial.

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

- This simplifies to \(\frac{x^2}{2} \cdot \frac{1}{x} = \frac{x}{2}\), which is a polynomial. While the multiplication does yield a polynomial, one of the factors here \(1/x\) is not a polynomial.
- Result: This does not represent a polynomial because \(1/x\) is not a polynomial.

D. \(x^2 + 2x + 1\)

- This expression is clearly a polynomial.
- Result: This represents a polynomial.

From these evaluations, the correct response that illustrates that polynomials form a closed system under multiplication is:

**A**: \((x^2+1)(x-1/2)\)

Though D is a polynomial itself, it doesn’t reflect the multiplication of two polynomials. Therefore, A clearly demonstrates the closure property.