Find completion of polynomial equipped with norm

Question

Let $\mathcal{P}$ denote the space of polynomials with respect to the norm $$\| a_0 + a_1 x + \dots + a_nx^n\| = |a_0| + |a_1| + \dots + |a_n|.$$ Determine the completion of $\mathcal{P}$.

I've never seen an example actually computing the completion of something, so I really don't know where to start

Thanks

Answer 1

Modification of Theo Bendit's answer:

Hint: show that this space is isometrically isomorphic to c00, the space of real sequences with finite support, under the $\Vert \Vert_1$ norm, where $\Vert a \Vert_1:=\sum_n \vert a_n\vert$. Then find a Banach Space in which c00 is a dense subspace.