i think so. in general, if M is a maximal ideal in a commutative ring R, then R/M is a field. If R contains a subfield k, then R/M is an extension of k. So in your case the ring K[X] contains the subfield K, and since K[X] is a Euclidean ring, hence also a p.i.d., an irreducible polynomial f generates a maximal ideal, so K[X]/(f) is a field extension of K.
Moreover the degree (vector dimension) of the extension equals the degree of f, hence is finite, and every finite extension is definitely algebraic. so YES!
I had to think through all the details since I am old and losing my memory. hope this helps.