Prove Coefficients of Complex Polynomials are Real

[de1irious]

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!

 

[gel]

Try a Taylor series about 0.

 

[mrandersdk]

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*