If $K/F$ is a finite extension of fields what is $K \otimes_F F[X]$?

Question

Suppose that $K/F$ is a finite extension of fields with $[K:F] = n$. Then since $K$ is an $n$-dimensional $F$ vector space, we have $K \cong \bigoplus_{i=1}^n F$. Then \begin{align*} K \otimes_F F[X] &\cong \bigoplus_{i=1}^n F \otimes_F F[X] \\ & \cong \bigoplus_{i=1}^n (F \otimes_F F[X]) \\ & \cong \bigoplus_{i=1}^n F[X]. \end{align*} Is this right? For some reason I feel like I'm missing something. In particular, the way I used the assumption $[K:F] = n$ is bothering me. Also, $F[X]$ is just sitting there and could easily have been any $F$ module.