The vector space of polynomials with degree at most $n$, $P_n$, has the basis \begin{equation*} B=\set{1,\,x,\,x^2,\,x^3,\,\ldots,\,x^n}. \end{equation*}
Another nice basis for $P_n$ is \begin{equation*} C=\set{1,\,1+x,\,1+x+x^2,\,1+x+x^2+x^3,\,\ldots,\,1+x+x^2+x^3+\cdots+x^n}. \end{equation*}
Checking that each of $B$ and $C$ is a linearly independent spanning set are good exercises.