Each district may be divided into as many divisions as there are directors of the district. The establishment of the divisions and the boundaries thereof shall be made by the board of supervisors, who shall make such division if requested by a resolution passed by a majority of the board of directors or upon a petition signed by a majority of eligible voters within the boundaries of the district. Said resolution or petition shall be filed with the board of supervisors at least 140 days before the next general district election.