The Degree of Zero Polynomial.

Question

I wonder why the degree of the zero polynomial is $-\infty$ ?

I heard that, it is $-\infty$ to make the formula $\deg(fg)=\deg(f)+\deg(g)$ hold when one of these polynomials is zero. However, if that was the only reason we could have said that it is $\infty$ instead.

Answer 1

One also wants $\deg(P+Q)\leq\max(\deg P,\deg Q)$ to hold, even if $P=-Q$.

Added much later: and maybe more importantly, in Euclidean division of some polynomial $A$ by $B\neq0$ we want the remainder $R$ to satisfy $\deg(R)<\deg(B)$, even if the division is exact (i.e., if $R=0$).