Inequality $\deg(f-g) \le \max(\deg(f),\deg(g))$

Question

$f, g$ polynomials over $\mathbb{C}$. In my book written that $\deg(f-g) \le \max(\deg(f),\deg(g))$ is FALSE. Why? I can't find any counterexample, maybe somebody can explain?

Answer 1

.It is possible that you might be missing something. Indeed, the inequality as given is true, since for any $i \geq 0$, the coefficient of $x^i$ in $f-g$ being non-zero implies that the coefficient of $x^i$ in one of $f$ or $g$ has to be non-zero, by the fact that $x + y \neq 0 \implies $ at least one of $x,y \neq 0$ for $x,y \in \mathbb C$.

In fact, if $\deg f \neq \deg g$, then equality holds in the given statement.