Question about polynomial multiplication

Question

Let $f=\sum a_{j}x^{j}$ and $g=\sum b_{j}x^{j}$ be two nonzero polynomials with coefficients in some field. Fix a natural number $i \in [0, \deg f]$. I want to know if the following is true:

There exists $n\in [i, i+deg(g)]$ such that the coefficient $c_{n}=\sum_{k=0}^{n} a_{k}b_{n-k}$ of $fg$ is nonzero.

Note: this question is NOT asking whether $fg=0$.

Answer 1

$$ (x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1)(x-1) = x^9-1 $$ Let your $i$ be, for example, $3$.

Answer 2

Why, yes, since for any polynomial ring over an integral domain, the leading term of $fg$ is the product of the leading terms of $f$ and $g$, and hence $\deg(fg)=\deg f+\deg g$.