Vector space/Polynomials/Short/Example

Let ${}R=K[X]$ be the polynomial ring in one variable over the field ${}K$, consisting of all polynomials, that is expressions of the form

$a_{n}X^{n}+a_{n-1}X^{n-1}+\cdots +a_{2}X^{2}+a_{1}X+a_{0},$

with ${}a_{i}\in K$. Using componentwise addition and componentwise multiplication with a scalar ${}s\in K$ (this is also multiplication with the constant polynomial ${}s$), the polynomial ring is a ${}K$-vector space.