Answer: No, the set of rational expressions is not closed under division. To see why, consider the following counterexample:
Let p(x) = x + 1, q(x) = x - 1, r(x) = x + 2, and s(x) = x - 2. Then,
p(x)q(x) = (x+1)(x-1) = x^2 - 1
r(x)s(x) = (x+2)(x-2) = x^2 - 4
So,
p(x)q(x) ÷ r(x)s(x) = (x^2 - 1) / (x^2 - 4)
This expression is a rational expression, but it is not defined for x = 2 or x = -2, since the denominator becomes zero at those values. Therefore, the set of rational expressions is not closed under division.
Explanation: