Recall the definition of a polynomial expression. Find two polynomial expressions whose quotient, when simplified, is . Use that
division problem to determine whether polynomials are closed under division. Then describe how the other three operations—addition, subtraction, and multiplication—are different from division of polynomials.
Polynomials are not closed under division. Many examples can prove that they are not. Any division of polynomials that leaves a variable to the first or higher power in the denominator is not a polynomial because it has a variable with an exponent that is not a positive integer.
When we add or subtract two polynomials, we’re combining like terms, or terms with the same power of the same variable. Therefore, the exponents and variables don’t change. Only the coefficients of each term might change. This means that the result must be a polynomial, and therefore polynomials are closed under subtraction.
The product of polynomials can have the same original variables as the factors but with higher integer exponents. These products will also be polynomials.
