Polynomial ring/Ordered field/Positivity/Exercise

Let ${}K$ be an ordered field and let ${}R=K[X]$ be the polynomial ring over ${}K$. Let

${}P={\left\{F\in K[X]\mid {\text{The leading coefficient of }}F{\text{ is positive}}\right\}}\,.$

Show that ${}P$ fulfils the following three properties.

1. Either ${}F\in P$ or ${}-F\in P$ or ${}F=0$.
2. If ${}F,G\in P$, then also ${}F+G\in P$.
3. If ${}F,G\in P$, then also ${}F\cdot G\in P$.

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