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Complex polynomials

  1. Feb 7, 2008 #1
    I need to show that the coefficients of a complex polynomial P(z) are real iff P(x) is real for all real x. Thanks!
     
  2. jcsd
  3. Feb 7, 2008 #2

    gel

    User Avatar

    Try a Taylor series about 0.
     
  4. Feb 8, 2008 #3
    try to look at P(x) and p(x)* where x is real (what do you know about z and z* if they are real). And write

    a + b x + c x^2 + ... d x^n

    and

    (a + b x + c x^2 + ... d x^n)*

    and compare them exploiting that x and p(x) is real. If this isn't enough i have made a spoiler in the bottom of the post, just follow the dots.
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    a + b x + c x^2 + ... d x^n = real = real* = (a + b x + c x^2 + ... + d x^n)* = a* + b* x* + c* x*^2 + ... + d* x*^n = a* + b* x + c* x^2 + ... + d* x^n

    so

    0 = (a-a*) + (b-b*)x + (c-c*)x^2 + ... + (d-d*)x^n for all x

    so

    a=a*, b=b*, c=c* ... d=d*
     
    Last edited: Feb 8, 2008
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