Axiom 1: Clearly the sum of two rational functions Is again a rational function.

Axiom 2: If p(x) is a polynomial, then so is op(x) where @ is a real number.

Hence \(\displaystyle{a}{\left({}\frac{{{p(x)}}}{{{q}{\left({x}\right)}}}\right)}\) is a rational function for all real a.

Axiom 3: Becatise addition of functions is defined pointwise, this follows from commutativity of addition of real numbers.

Axiom 4: Same for associativity.

Axiom 5: Notice that the zero function \((x) = 0 \) is « rational function because it is a polynomial whose every coefficient is 0.

Axiom 6: Follows from existence of negatives in the reals.

Axiom 7: Follows from associativity of multiplication of the reals.

Axiom 8: Follows from distributive law of the reals.

Axiom 9: Also follows from the distributive law of the reals.

Axiom 10: \(\displaystyle{1}{\left({}\frac{{{p(x)}}}{{{q}{\left({x}\right)}}}\right)}=\frac{{{p}{\left({x}\right)}}}{{{q}{\left({x}\right)}}}\) for all polynomials p and q.