In one my question, a guy told me about the long divison of polynomials and said that-
"Given any polynomials $f$ and $g$, there exist polynomials $q$ (the quotient) and $r$ (remainder) such that $$ f=q⋅g+r $$
and the degree of $r$ is strictly smaller than the degree of $g$."
I understand that in divison the reminder is smaller than the divisor but is it necessary for the remainder to be smaller in degree than the divisor?
I mean for example if the divisor was $2x^2$ then is it possible for the remainder to be $x^2$ or does it have to be in the degree of $1$ or smaller?
Is it possible for the remainder to be the same degree as the divisor but still be smaller than it.