Good question!

Yes, and in fact, if we're being careful, probably the best way to define the "constant term of a polynomial" would be to say it is the coefficient of the $x^0$ term.

As a side note, this is somewhat the reason that the degree of the "zero polynomial" is not defined using our usual definition. For example

• $f(x) = x^2 + 3x + 4$ has degree $2$ because the $x^2$ term is the highest power with a non-zero coefficient (which is $1$). No problems here.
• $f(x) = 5$ has degree $0$ because the $x^0$ term is the highest power with a non-zero coefficient (which is $5$). Still no problem here.
• What about $f(x) = 0$? Here there is no $x^0$, $x^1$, $x^2$, etc. term that has a non-zero coefficient, which is why our usual definition of degree does not work.