division
None.
Yes, because there is no way of multiplying two polynomials to get something that isn't a polynomial.
The set of rational numbers is closed under all 4 basic operations.
You don't say that "an integer is closed". It is the SET of integers which is closed UNDER A SPECIFIC OPERATION. For example, the SET of integers is closed under the operations of addition and multiplication. That means that an addition of two members of the set (two integers in this case) will again give you a member of the set (an integer in this case).
division
It means that you can do any of those operations, and again get a number from the set - in this case, a polynomial. Note that if you divide a polynomial by another polynomial, you will NOT always get a polynomial, so the set of polynomials is not closed under division.
None.
Yes, because there is no way of multiplying two polynomials to get something that isn't a polynomial.
Yes. The empty set is closed under the two operations.
Yes.
The set of rational numbers is closed under all 4 basic operations.
In my opinion the question is poorly defined, since "non-polynomial" could be just about anything.
you dont
Yes. When you add any whole numbers you get another whole number. That is what closed means in this context. The answer is still a whole number.
Polynomials are the simplest class of mathematical expressions. The expression is constructed from variables and constants, using only the operations of addition, subtraction, multiplication and non-negative integer exponents.
You don't say that "an integer is closed". It is the SET of integers which is closed UNDER A SPECIFIC OPERATION. For example, the SET of integers is closed under the operations of addition and multiplication. That means that an addition of two members of the set (two integers in this case) will again give you a member of the set (an integer in this case).