Yes, because there is no way of multiplying two polynomials to get something that isn't a polynomial.
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Yes.natural numbers are closed under multiplication.It means when the operation is done with natural numbers in multiplication the sum of two numbers is always the natural number.
The set of rational numbers is closed under all 4 basic operations.
Yes. The set of real numbers is closed under addition, subtraction, multiplication. The set of real numbers without zero is closed under division.
If you can never, by multiplying two whole numbers, get anything but another whole number back as your answer, then, YES, the set of whole numbers must be closed under multiplication.