Final answer:
Relations between the Pope and Patriarch were weakened by disputes over religious images, church authority, and the Pope's political actions, particularly the crowning of Charlemagne, leading to the Great Schism of 1054.
Step-by-step explanation:
The factors that weakened the relations between the Pope and the Patriarch included the argument over icons, the Patriarch's refusal to recognize the Pope as the head of the Church, and the Pope's crowning of Charlemagne as emperor. The iconoclastic controversy caused a rift as the eastern and western churches debated over the use of religious images. The eastern church, administered from Constantinople, allowed marriage for priests and conducted religious services in Greek, while the western church, led by the Pope in Rome, insisted on a celibate clergy and used Latin. The crowning of Charlemagne by Pope Leo III as Holy Roman Emperor exacerbated the rift by signaling the Pope's ambition to assert temporal authority over Christian Europe, a move which the eastern church saw as undermining their religious and political autonomy.
A definitive split known as the Great Schism of 1054 occurred after mutual excommunications were declared by both leaders, solidifying the division into what would become the Eastern Orthodox Church and the Roman Catholic Church. Over time, doctrinal differences, language barriers, cultural distinctions, and political disputes continued to strain the relationship between the two branches of Christianity.