If your network is large enough to need 8 routers I wonder if the capacity of a cable modem will really be sufficient for internet connectivity.
But you did not ask if it would work well, you asked if it would work. The best way to make it work is to run a dynamic routing protocol among your 8 routers and to have router A announce a default route to the others. The mechanics of how to do that depend on which routing protocol you choose, but for example with OSPF you configure default-information originate. This will provide that the other routers will forward packets for unknown destinations to router A.
On router A you would configure a static default route pointing to the interface with the cable modem. You would also configure on the interface with the cable modem address negotiated so that the router will negotiate with the cable modem and accept the address that the cable modem assigns.
HTH
Rick
HTH
Rick