the teacher together with some students are visiting the fac

应该改为 The teacher together with some students 【is】 visiting the factory.
本句的主语是The teacher,单数.如果把together with 换成 and ,主语就是复数概念了.
注意：当主语是用with,together with,as well as,like,besides,except,but等连接的两个词时,谓语动词与第一个词一致,即“就前一致”.如 Tom with his classmates is playing outside./ Mr.John as well as his students is talking in the office.