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Degree of a Constant

Date: 12/02/2002 at 14:10:12
From: Joy Sapienza
Subject: Degree of a constant

Dear Dr. Math,

Why is the degree of a constant zero?

We have just finished studying that any number to the zero power is 
one.

It makes sense that the degree of a monomial like 2x is one, but it 
doesn't make sense to me that in a constant like 5, the degree is 
zero.

Please help.
Thank you,
Joy


Date: 12/02/2002 at 14:41:35
From: Doctor Ian
Subject: Re: Degree of a constant

Hi Joy,

It looks like you mostly figured it out yourself. Each term in a
polynomial is the product of a constant (which may be 1 or 0) and a
variable raised to an exponent, e.g., 

  3x^4 + 0x^3 + 1x^2 + 5x^1 + 7x^0

For convenience, we leave out terms where the coefficient is zero, 

  3x^4 + 1x^2 + 5x^1 + 7x^0

and we leave multiplication by 1 implied, 

  3x^4 + x^2 + 5x^1 + 7x^0

We abbreviate x^1 as just x, 

  3x^4 + x^2 + 5x + 7x^0

and we simplify x^0 to 1... which is left implied (see above):

  3x^4 + x^2 + 5x + 7

So it's convenient to write a polynomial this way, but it's much
cleaner for the definition to require no special cases:

           n
         ---
         \       i
  p(x) = /    a x
         ---   i
          i=0

When n=0, we get

      0
   a x  = a
    0      0

When you look at it this way, a constant is just a very short
polynomial. This is nice, because you can just have one set of rules
for dealing with polynomials of any degree, and constants are covered
by it.  

Does this make sense? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Polynomials

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