Degree of Zero

Date: 02/25/2003 at 13:02:28
From: Kristi
Subject: Degree of zero

Dear Dr. Math,

I just wanted to know what the degree of zero was.

Thank you.

Date: 02/25/2003 at 20:19:28
From: Doctor Rick
Subject: Re: Degree of zero

Hi, Kristi.

That's an interesting question. I assume you're asking for the degree 
of the function f(x) = 0 regarded as a polynomial. In general, a 
constant function is regarded as a polynomial of degree zero, as we 
discuss here:

   Degree of a Constant
   http://mathforum.org/library/drmath/view/61845.html 

   Degree of Constant Function
   http://mathforum.org/library/drmath/view/54602.html 

This is true because a constant such as 2 can be regarded as 2*x^0, 
and the degree of a polynomial is the highest power of the variable 
that has a non-zero coefficient.

I presume this is why you asked the question - there is no term in 
f(x) = 0 that has a non-zero coefficient. So do we say that this 
function has no degree?

To me, yes, this is the most sensible answer. Think about degree 
another way: a polynomial has (at most) as many zeros as its degree. 
(It has exactly as many zeros as its degree if we count the 
multiplicities of the zeros, for instance, x^3-10x^2+33x-36 = 
(x-3)^2(x-4) has 3 zeros: 3, of multiplicity 2, and 4, of multiplicity 
1.)

A non-zero constant function like f(x) = 5 has zero zeros, in keeping 
with its degree of zero. But f(x) = 0 has an infinite number of zeros: 
every value of x is a zero of the function. Thus it makes sense to say 
that the degree of f(x) = 0 is undefined.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/