Abstract dialectical frameworks (ADFs) are one of the most powerful generalizations of classical Dung-style argumentation frameworks (AFs). The additional expressive power comes with an increase in computational complexity, namely one level up in the polynomial hierarchy in comparison to their AF counterparts. However, there is one important subclass, so-called bipolar ADFs (BADFs) which are as complex as classical AFs while offering strictly more modeling capacities. This property makes BADFs very attractive from a knowledge representation point of view and is the main reason why this class has received much attention recently. The semantics of ADFs rely on the Gamma-operator which takes as an input a three-valued interpretation and returns a new one. However, in order to obtain the output the original definition requires to consider any two-valued completion of a given three-valued interpretation. In this paper we formally prove that in case of BADFs we may bypass the computationally intensive procedure via applying Kleene's three-valued logic K. We therefore introduce the so-called bipolar disjunctive normal form which is simply a disjunctive normal form where any used atom possesses either a positive or a negative polarity. We then show that: First, this normal form is expressive enough to represent any BADF and secondly, the computation can be done via Kleene's K instead of dealing with two-valued completions. Inspired by the main correspondence result we present some first experiments showing the computational benefit of using Kleene.