This question tests your understanding of participles and participle phrases used to show reason or cause. The blank needs a phrase that explains why Borussia Dortmund will go down to the second division. The reason is their performance: they were beaten in every match.

Let's analyze the options:

Step 1: Identify the needed grammatical form.
The phrase in the blank is describing the subject "Borussia Dortmund" and providing the reason for the main clause ("will go down"). This is the job of a participle phrase. Since the team received the action of being beaten (they did not do the beating), we need a passive construction.

Step 2: Evaluate the options.

• Beaten: This is a past participle and can be used passively. However, on its own, "Beaten in every match..." can sound slightly incomplete or like an imperative command ("Be beaten!"), which doesn't make sense here.
• Having beaten: This is an active perfect participle ("having" + past participle). It means the subject (Borussia Dortmund) did the beating to someone else. This is the opposite of the intended meaning.
• To be beaten: This is a passive infinitive. It often expresses purpose or something that is going to happen in the future (e.g., "They are ready to be beaten"). It does not effectively explain a reason that has already happened throughout the season.
• Having been beaten: This is a passive perfect participle. "Having been" + past participle is used to show an action that was completed in the past before the main action. This perfectly captures the meaning: Because they had been beaten in every match, they will now be relegated.

Final Answer: The most appropriate word is Having been beaten.

Related Topic: Participle Phrases

A participle phrase is an adjective phrase that begins with a present participle (-ing) or a past participle (-ed, -en, etc.). It modifies a noun or pronoun and often explains the cause, reason, or time of the main action in the sentence. The perfect form ("having" + past participle) emphasizes that the action of the participle phrase was completed before the action of the main verb.

Example (Active): Having finished her homework, she went to bed. (She finished her homework first, then went to bed.)

Example (Passive): Having been warned about the storm, we stayed indoors. (We were warned first, so we stayed indoors.)

Related Formulae (Grammar Rules)

1. Present Participle Phrase (Active): [verb]-ing + object...
    Function: Shows an action happening at the same time as the main verb.

2. Past Participle Phrase (Passive): [past participle]...
    Function: Describes a noun that received an action.

3. Perfect Participle Phrase (Active): Having + [past participle] + object...
    Function: Shows an action that was completed before the main action.

4. Perfect Participle Phrase (Passive): Having been + [past participle]...
    Function: Shows an action that was received and completed before the main action.