= Solution
For the <pointwise order on real polynomials>, $p(x)\le p(x)$ proves <reflexivity>. Two opposite inequalities imply $p(x)=q(x)$ for every real $x$; the <polynomials> are therefore equal, proving the <antisymmetric relation> axiom. Chaining the inequalities at every $x$ proves <transitivity>.
To disprove <total order>, take $p(x)=x$ and $q(x)=0$. The inequality $p\preceq q$ fails at $x=1$, whereas $q\preceq p$ fails at $x=-1$. \b[The order is partial but non-total.]
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