What is the degree of polynomial $0$?

Question

I was reading some handouts for my linear algebra course, and a note captured my attention.

The degree of the polynomial $0$ is usually either undefined, or it's defined as $-\infty$.

I am trying to make sense of this statement. Any constant polynomial $c$ is said to have $\deg = 0$. Why would it work any differently for $0$?

My conjecture is that, while a polynomial like $p(x) \equiv 1$ implies that the highest degree of $x$ is $0$ inside the polynomial, because it couldn't be otherwise, $0$ doesn't imply such a fact. In fact, the polynomial $q(x) \equiv0$ might very well be the expression $0x^{100} - 0x^2 + 0x$, hence no hypothesis can be made on the highest degree of the variable.

Does this make sense? And if so, is it really a good enough reason to have to make up a separate rule for the polynomial $0$ instead of treating it like any other constant one?

One reason I think of is the formula for the degree of the product of two polynomials which says that deg(pq) = deg(p) + deg(q) — Amadeus Maldonado, Feb 29 '20 at 11:41
The degree of a polynomial $P(x)$ is the power of $x$ in its highest-order non-zero term. — TonyK, Feb 29 '20 at 11:58

score 5 · Answer 1 · edited Feb 29 '20 at 11:54

5

Your explanation make sense.However,the following fact is deserved to mention. If the degree of polynomial $0$ is defined to be $-\infty$, the following formula is still true. $$\text{degree}(g\cdot f)=\text{degree}(g)+\text{degree}(f).$$

edited Feb 29 '20 at 11:54

Samuele B.

761

answered Feb 29 '20 at 11:44

Mr.xue

681

What is the degree of polynomial $0$?

1 Answers1