Is the quotient ring $K[x]/\langle f\rangle$ a $n$-dimensional $K$-vector space?

Question

Let $K$ be a field and $f(x)\in K[x]$ with $\deg(f)=n$.
Then, I have to proof that the quotient ring $K[x]/\langle f\rangle$ is a $K$-vector space of dimension $n$.
But I can't think of any scalar product to elements of the form $p(x)+\langle f\rangle$ with elements of $K$ (Actually, I thought of the elements of $K$ as the constant polinomials $q(x)=k$, $\forall\ k\in K$, but I'm not so sure of it).
Also, I remembered a hint about proposing the set $\{1,x,x^2,\dots,x^{n}\}$ as the base for the ring, but I think it's wrong since it must be until $x^{n-1}$ instead of $x^n$.