This is only a matter of semantics.

Sometimes we say

p(x) is a polynomial p in the variable x. 

and other times we say

p(x) is a polynomial p in x. 

It's your choice to either put in the words "the variable" or leave them out.

Ask your teacher which one he or she prefers.

But, as I said above, it's just a matter of preference which you say.
It makes no difference whatever mathematically speaking.

Edwin