splitting coax cable to cable box and modem/router

hello, im hoping someone could help me.  i think im on the right path and this site looks to be the place to get the best answers.  here is my situation.  my xfinity modem/router is upstairs and i would like to bring it downstairs to my den where my main cable box is.  there is only 1 coax "outlet" (not sure if that is the correct word) down here.  currently it goes directly into my cable box.  i want to bring the modem/router down here so i can run a landling/ethernet cable directly to my ps4.  the download speeds over wifi is tremdously slow.  the question.  can i just get a splitter (i probably have an old one) and split the cable from the wall to go to my cable box and my modem/router then just plug an ethernet cable from the modem/router to my ps4.  will this work? will it slow anything down? should it speed up my download speed to my ps4?  not to mention by bringing the modem/router downstairs it will be 4 floors to the bedroom, i bought some wifi boosters to help get the signal upstairs, just waiting on them to arrive.  thanks in advance.