The English translation “factious” is just a poor translation. The implications of the word differ from the Greek. Here is the usage described in the TDNT:
From this there develops in Hellenism the predominant objective use of the term to denote a. “doctrine” and especially b. “school.” The αἵρεσις of the philosopher, which in antiquity always includes the choice of a distinctive Bios, is related to δόγματα to which others give their πρόσκλισις. It thus comes to be the αἵρεσις (teaching) of a particular αἵρεσις (school).1 Cf. the title of a work by Antipater of Tarsus (2nd century B.C.) κατὰ τῶν αἱρέσεων, and the writing of Chrysipp. αἵρεσις πρὸς Γοργιππίδην (Diog. L., VII, 191); also the description of the philosophical schools as αἱρέσεις in Polyb., V, 93, 8 (Peripatetic), Dion. Hal. Compos. Verb., 19, p. 134, 3 f. (ἥ γʼ Ἰσοκράτους καὶ τῶν ἐκείνῳ γνωρίμων αἵρεσις); Sext. Emp. Pyrrh. Hyp., 1, 16; Diog. L., I, 19 (τοῦ δὲ ἠθικοῦ [sc. μέρους τῆς φιλοσοφίας] γεγόνασιν αἱρέσεις δέκα: Ἀκαδημαική, Κυρηναική κτλ.). For the concept of such a fellowship—as well as αἱρέσεις κατὰ φιλοσοφίαν (Sext. Emp. Pyrrh. Hyp., I, 185) we also have κατὰ ἰατρικὴν αἱρέσεις (ibid., I, 237)—the following aspects are important: the gathering of the αἵρεσις from a comprehensive society and therefore its delimitation from other schools; the self-chosen authority of a teacher; the relatively authoritarian and relatively disputable doctrine; and the private character of all these features.
We need not be surprised if in Philo it is used on the one side to denote a Greek philosophical school, as, e.g., in Plant., 151, and if on the other it is employed to depict what Philo calls the august philosophical society of the Therapeutics, as, e.g., in Vit. Cont., 29. In Josephus, too, αἵρεσις is used of the religious community of the Essenes (Bell., 2, 118). Indeed, Josephus sees all the Jewish religious schools in terms of the Greek philosophical schools, the Essenes, Sadducees and Pharisees being the τρεῖς παρʼ ἡμῶν αἱρέσεις. After his investigation3 of all three, Josephus resolved πολιτεύεσθαι τῇ τῶν Φαρισαιων αἱρέσει κατακολουθῶν, ἣ παραπλήσιός ἐστι τῇ παρʼ Ἕλλησιν Στωϊκῇ λεγομένῃ (Vit., 12).
Heinrich Schlier, “Αἱρέομαι, Αἵρεσις, Αἱρετικός, Αἱρετίζω, Διαιρέω, Διαίρεσις,” ed. Gerhard Kittel, Geoffrey W. Bromiley, and Gerhard Friedrich, Theological Dictionary of the New Testament (Grand Rapids, MI: Eerdmans, 1964–), 181. The underlying verb simply to grasp, take, et cetera. It is someone creating a group. Perhaps the best analogy to Acts 3:10 is the savage wolf of Acts 20:29. Such a group would have the secondary effect of division, but division is not issue. It certainly does not mean disagreement over finances, et cetera.