Verified answer

yes, but the term containing zero as its numerical coefficient doesn't have any value... rule of zero applies... :)

You can have. But if there is a term with zero as coefficient, then the expression will be the same even if you remove these terms. Therefore,

(x^7+0x^5-0x^2+19)/(x-1) is the same as

(x^7+19)/(x-1)

yep yep

possible..

but the equation will be reduced to,

x^7+19 times the whole quantity x-1

Yes, but you have to cancel everything that has a zero in it.

that's possible. synthetic division uses that so that we can pretend that there is a term existing in decreasing orders for easier division.