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Polynomial practice 2 (Posted on 2020-11-04) Difficulty: 3 of 5
Determine all polynomials P such that for every real number x, P(x)2+P(-x)= P(x2)+P(x)

No Solution Yet Submitted by Danish Ahmed Khan    
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Some Thoughts Solution? Comment 1 of 1
P(x) = x^n for any even power works 
Thus includes the constant function P(x)=1
Also the zero polynomial P(x)=0

It's easy to show x^n does not work for odd powers.

If the polynomial has more terms the LHS will have terms with powers lacking on the RHS.  These come from P(x)^2 compared to P(x^2).  Furthermore, these extra terms are of higher degree than P(x) so there's no way to fix this with the difference between P(-x) and P(x). 

  Posted by Jer on 2020-11-04 10:45:16
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