How would do this Algebra question?

Question

$(b)$ Let $\mathbb{F}$ be a field and $X$ an indeterminate, and consider the polynomial ring $\mathbb{F}[X].$

$\quad(\mathsf{i})$ Let $f(X),g(X)\in\mathbb{F}[X]$ with $f(X),g(X)\neq0.$ Prove that $f(X)g(X)\neq0.$

$\quad(\mathsf{ii})$ Is $\mathbb{F}[X]$ an integral domain? Justify your answer. You may assume that $\mathbb{F}[X]$ is a commutative ring.

How would I answer each part of this question, it makes no sense?

Answer 1

i) Consider the leading coefficient of $f(x)g(x)$

ii) Yes, by the point i)