To determine whether polynomials form a closed system under multiplication, we need to verify if the product of any two polynomials is also a polynomial.
Let's analyze the provided options to see if they represent the product of two polynomials:
-
(x² + 1)(x − 1):
- Here, both \(x² + 1\) and \(x - 1\) are polynomials. The product of two polynomials is given by: \[ (x^2 + 1)(x - 1) = x^3 - x^2 + x - 1 \] This result is also a polynomial, demonstrating closure under multiplication.
-
(√√√x)(x + 1):
- The term \(\sqrt[3]{x}\) is not a polynomial, as it can be expressed as \(x^{1/3}\), which has a non-integer exponent. Therefore, this expression does not demonstrate closure, as it includes a non-polynomial factor.
-
(꽃) (금):
- Since these terms are not polynomial expressions (they appear to be characters rather than algebraic expressions), we cannot assess the closure under multiplication meaningfully.
-
2 (a constant):
- A constant, such as 2, is considered a polynomial of degree 0. However, multiplying it by another polynomial (like any polynomial given) still results in a polynomial. While this demonstrates closure when combined with other polynomials, it isn't a product of two non-constant polynomials.
Given this analysis:
The response that clearly demonstrates that polynomials form a closed system under multiplication is the option (x² + 1)(x − 1).
1 answer