
Zero Polynomial problem
Let $\displaystyle A \in R[x] $ have degree $\displaystyle n \geq 0 $ and let $\displaystyle B \in R[x] $ with degree of at least 1 with the leading coefficient of B being a unit of R. Then by a problem I posted earlier, we know that there exist unique polynomials $\displaystyle Q_0,Q_1,...Q_n \in R[x]$ such that $\displaystyle deg Q_i < deg B \ \ \ \forall i $. Prove that if $\displaystyle deg(B)=1 $, then $\displaystyle Q_n$ is not the zero polynomial.
Idea so far.
So if the degree of B is 1, then the degree of $\displaystyle Q_n$ is less than 1, so does that force it to be a constant polynomial that is not 0 since the leading coefficient is a unit?
Thank you.