whole numbers are closed under addition because the sum of two whole numbers is always a whole number…

Brief Explanations:

When checking polynomial division using the formula (dividend = divisor\times quotient+remainder). For multiplication, if we consider two polynomials as divisor and quotient, their product (part of the right - hand side) is a polynomial. For addition, the sum of the product and the remainder (if any) is also a polynomial. This shows that when we perform multiplication and addition operations on polynomials within the context of division - checking, the result is always a polynomial, thus supporting closure under these operations.

Answer:

The process of checking polynomial division (dividend = divisor\times quotient + remainder) shows that the product of polynomials (divisor and quotient) is a polynomial (multiplication closure) and the sum of this product and the remainder (if any) is a polynomial (addition closure).