Whole numbers are closed under addition because the sum of two whole numbers is always a whole number. Explain how the process of checking polynomial division supports the fact that polynomials are closed under multiplication and addition.

The correct answer is:

When checking polynomial division, you multiply the quotient by the divisor.

The quotient will be a polynomial (with or without a remainder). Multiplying this polynomial by the polynomial divisor, we get a polynomial in which the exponents and coefficients have changed. Thus polynomials are closed under multiplication.

There will usually be at least two terms in the divisor. When we multiply the quotient by this, we will use the distributive property, multiplying the entire quotient by each term of the divisor. When this process is finished, we will need to add the polynomials we had from multiplying together. Doing this, we get a polynomial answer in which the coefficients have changed. Thus polynomials are closed under addition.

A polynomial is a finite sum of terms in which all variables have whole number exponents and no variable appears in a denominator.

When adding polynomials, the variables and their exponents do not change. Only their coefficients will possibly change. This guarantees that the sum has variables and exponents which are already classified as belonging to polynomials. Polynomials are closed under addition.

When multiplying polynomials, the variables' exponents are added, according to the rules of exponents. Remember that the exponents in polynomials are whole numbers. The whole numbers are closed under addition, which guarantees that the new exponents will be whole numbers. Consequently, polynomials are closed under multiplication.