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Degree of the Zero polynomial

  1. Jul 20, 2012 #1
    I understand that mathematicians have had to define the number '0' also as a polynomial because it acts as the additive identity for the additive group of poly's.What I do not understand is why they define the degree of the zero polynomial as [ tex ]-\infty[ /tex ].

    An explanation on planetMath wasn't that helpful,at the end they point-out to refer to the extended real numbers(don't they mean 'projectively extended real numbers??)
     
    Last edited: Jul 20, 2012
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  3. Jul 20, 2012 #2

    Mute

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    Well, I guess it's similar to how one sometimes regards zero as both a real and an imaginary number because you can write 0 = 0 + i0. Similarly, you can write

    0 = 0 + 0x + 0x2 + 0x3 + ...

    i.e., you can write '0' as an infinite degree polynomial with all coefficients zero.

    (There may be a more rigorous reason, but that's an intuitive one).
     
  4. Jul 20, 2012 #3
    Thanks!Could you explain what extended real numbers have got to do with this?
    But,polynomials always have non-negative degrees.
    deg[P(x)]=+n
    Why would mathematicians define a polynomial with a negative degree?
     
  5. Jul 20, 2012 #4

    eumyang

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    Huh, I'm looking at a precalculus textbook (Larson, 8th Ed.), and it states that the zero polynomial has no degree. Is that wrong? (Note that no degree ≠ zero degree -- a polynomial that consists of a single non-zero number has a degree of zero.)
     
  6. Jul 20, 2012 #5

    micromass

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    The choice is pretty arbitrary. Sometimes it defined as having no degree, sometimes it's -1, sometimes it's [itex]-\infty[/itex].

    A handy formula for polynomials is

    [tex]deg(P)+deg(Q)=deg(PQ)[/tex]

    If we want this formula to hold for the zero polynomial, then we see (by taking Q=0) that

    [tex]deg(P)+deg(0)=deg(0)[/tex]

    must hold for all P. This is only satisfied with [itex]deg(0)=-\infty[/itex]. This is the reason why they defined it this way. But again, it's pretty arbitrary.
     
  7. Jul 20, 2012 #6

    eumyang

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    I see now. Thank you.
     
  8. Jul 20, 2012 #7
    Awesome!Thanks a ton.
     
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