Why is the zero polynomial not assigned a degree? Yesterday, I read in my textbook,

We assign degree to every polynomial and even a non-zero constant is assigned a degree $0$ but $0$ itself is not assigned a degree.

Why is that? Why we don't assign degree $0$ to the zero polynomial?
 A: This is to make nice rules such as
$$
\text{deg }(PQ) = \text{deg }P + \text{deg }Q\\
\text{deg }(P+Q) \le \max(\text{deg }P , \text{deg }Q)
$$
So the only value that makes it possible is
$$\text{deg }0= -\infty$$
A: The degree of a polynomial is the exponent of the term with the highest power (and non-zero coefficient).
$$5x-4x^4+2\to x^4\to 4$$
$$0x^{45}+1x^3+2x^2\to x^3\to3$$
$$21\to x^0\to0$$
$$0\to ??$$
The null polynomial contains no power of the variable.
A: Assigning a degree to the zero polynomial will cause trouble with important and useful theorems that relate the degree of a polynomial to its roots:
If $F$ is a field (examples of fields are $\mathbb{R}$, $\mathbb{C}$, $\mathbb{Q}$, $\mathbb{Z}/p\mathbb{Z}$), a polynomial $P$ with coefficients in $F$ (the set/ring of these polynomials is usually denoted by $F[x]$) of degree $n$, has at most $n$ distinct points $\alpha\in F$ such that $P(\alpha)=0$.
This theorem follows from the fact that we can repeatedly factor out terms of the form $x-\alpha$ (where $\alpha$ is a root) from $P(x)$, lowering the degree of the remaining polynomial by $1$ in each step. See also: http://en.wikipedia.org/wiki/Factor_theorem.
When we restrict to polynomials with coefficients in $\mathbb{C}$, the statement is related to the Fundamental Theorem of Algebra.
Now since the zero polynomial in $F[x]$ has a root at every point in $F$, at least for infinite fields ($\mathbb{R},\mathbb{C},\mathbb{Q}$), we cannot assign a finite value to the degree of the zero polynomial without getting into trouble.
