Question

a. Show that the system of rational expressions is closed under addition, or give an example showing the contrary.

Answer 1

A "rational expression" is a fraction in which the numerator and the denominator are both polynomials. The sum of two such expressions will always produce another such expression, viz.,
p(x)/q(x) + r(x)/s(x) = [p(x)s(x) + q(x)r(x)] / [q(x)*s(x)].
The numerator and denominator of the sum will both be polynomials when fully expanded