To find the answer to this question, we need to analyze each answer choice and determine which one was a major factor in the decline of the Byzantine Empire.
A. The decision to divide the Roman Empire into two: While the division of the Roman Empire into the Western Roman Empire and Eastern Roman Empire did contribute to the divergence of their political and cultural paths, it was not a major factor in the decline of the Byzantine Empire.
B. The dispute between the pope and the patriarch of Constantinople: The Great Schism of 1054 created a permanent rift between the Western Roman Catholic Church and the Eastern Orthodox Church, but this dispute was not a primary factor in the decline of the Byzantine Empire.
C. Repeated attacks by Muslim invaders: This answer choice is a major factor in the decline of the Byzantine Empire. The Byzantines faced multiple invasions and attacks by various Muslim powers, such as the Arab caliphates and the Seljuk Turks, which significantly weakened the empire over time.
D. The conquests of Justinian in the west: While the conquests led by Emperor Justinian in the west, such as the reconquest of North Africa and Italy, briefly expanded the reach of the Byzantine Empire, it did not directly contribute to its decline.
Based on the analysis, the correct answer is C. Repeated attacks by Muslim invaders, as this was a major factor in the decline of the Byzantine Empire.