Degree of ZeroDate: 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/ |
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