Matthew 10:34 - 36
NIV - 34 Do not suppose that I have come to bring peace to the earth. I did not come to bring peace, but a sword. 35 For I have come to turn " 'a man against his father, a daughter against her mother, a daughter-in-law against her mother-in-law -
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I would say that it stems from Jesus' rightful claim to be of first importance (even above the closest earthly ties of family, friendship, or social/political allegiance) in the lives of His followers, as He indicated in passages such as Matthew 10:37-38 (immediately following the verses cited in the question). Conflict and division are sure to arise when people determine to follow Christ, and those who are close to them, or who believe that they have a greater right to the person's loyalties, cannot (because of their pride) understand or accept the change that this produces in the person's priorities. That pride is also the fundamental reason why the separation occurs to begin with -- because, apart from conviction by the Holy Spirit, people do not want to believe or admit that they are utterly incapable of saving themselves, and thus require a Savior. What is truly "good news" to the redeemed thus becomes offensive to the unsaved, and is such a fundamental difference in perspective that it leads to separation and estrangement.
I humbly submit for your consideration a very simple answer; that on our little sphere nothing will cause a quicker divide than the "TRUTH" on any given matter. (Psalms 145:18; 1 Corinthians 7:23; Galatians 4:8; Galatians 5:7) Some may even call [TRUTH] various fashionable idealogical names so as to confuse seeking to add credence or power for their weak position on the matter at hand. And still others demand protection via safe spaces to protect their fragile egg shell minds from the realities of any TRUTH; not of their own making. (2 Corinthians 11:15) Today "TRUTH" is mislabeled and considered a microaggression or worse and must be controlled via the sentimental marxism of political correctness. (Galatians 4:16) “He who marries the spirit of the age will soon be a widower.” Dean W. R. Inge In the Lord's freedom.....warrior on
The Gospel means "Good News", so why does good news divide, people? The verses you cited have been used by others to say Christianity isn't a religion of peace but Jesus is saying it brings conflict and division because of opposition and like all Scripture, it must be read in context. When Jesus came as the Messiah chaos did break out. Why?:  Satan's Reign as the Prince of the Air (Ephesians 2:1-2) in the world was immediately challenged and Satan tried to immediately destroy Jesus through his temptations in the wilderness, Matthew 4:1-11. War on!  Systematic religion as the power of bondage in the world was immediately challenged by Jesus. In John 14:6 Jesus says He is the only way to the Father, and the religious leaders immediately tried to destroy Jesus through lies and even death, Matthew 12:14. War on!  Social Rites as the pattern of structure in the world was immediately challenged by Jesus stating that clinging to your family and earthly passions had to be secondary in order to be His disciple, Matthew 16:24, Luke 9:24-25, and Luke 9:57-62. War on! When Jesus came everything changed and continues today. Satan, false religion and humanistic ritual all still challenge True Christianity, but the True Gospel or Good News remains the only way to real peace. Peace with God and Peace of God. The Word of God is a divider even of the soul and spirit, Hebrews 4:12. As a two-edged sword, so it divides self, society, and sacred ritual.
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