Question

Provide an example that shows the closure property for polynomials failing to work. (Think about what operation(s) were not included when you were learning about the closure property for polynomials.) Explain why your example does not show closure of polynomials.

Answer 1

An example that shows the closure property for polynomials failing to work is:

5x^2 + 2x + 1 / (2x - 1)

This fails to demonstrate the closure property for polynomials because polynomial division is not included in the basic arithmetic operations (addition, subtraction, multiplication) used when discussing the closure property for polynomials. Polynomial division requires a quotient, which is not necessarily a polynomial. For example, the quotient when performing the division above is:

2.5x + 3 / (2x -1) + 0.5

The 0.5 in the quotient is a constant term, not a polynomial, so the result of this division is not a polynomial. Therefore, polynomial division breaks the closure property of polynomials.

The closure property for polynomials states that when any two polynomials are combined using the basic arithmetic operations (addition, subtraction, multiplication), the result will always be a polynomial. Division is not one of these basic operations, so examples involving polynomial division, like the one shown, do not demonstrate closure of polynomials.