Question

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

(X) (1/2x)
(3x ^ 2)/(4 - 1)
1/(4x ^ 3) * 5
(sqrt(x))(6x * (- 3))

Answer 1

To determine whether polynomials form a closed system under multiplication, we need to confirm that the product of any two polynomials is also a polynomial.

Let's examine the provided responses one by one:

1. **(1/2)x**: This expression can be rewritten as \((1/2)x\), which is a polynomial of degree 1.
2. **(3x^2)/(4 - 1)**: Simplifying this gives \((3x^2)/(3) = x^2\), which is a polynomial of degree 2.
3. **1/(4x^3) * 5**: The term \(5/(4x^3)\) is not a polynomial, since it contains a term with a negative exponent when expressed as \(5 \cdot x^{-3}\).
4. **(sqrt(x))(6x * (-3))**: The term \((\sqrt{x})\) can be expressed as \(x^{1/2}\). The expression then becomes \((6x(-3))(x^{1/2}) = -18x^{3/2}\), which again is not a polynomial because of the fractional exponent.

From this analysis, the responses clearly show that not all expressions are polynomials and specifically that **(1/(4x^3) * 5)** and **(sqrt(x))(6x * (-3))** are not polynomials.

Therefore, polynomials do not form a closed system under multiplication because some products yield results that are not polynomials.