Find all polynomials P ( x ) with real coefficents satisfying P 2 ( x ) − 1 = 4 P ( x 2 − 4 x + 1 )

Question

Find all polynomials   P  (  x  ) with real coefficents satisfying       P    2    (  x  )  −  1  =  4  P  (      x    2    −  4  x  +  1  )

recepiamsb · Accepted Answer

Let   Q  (  x  +  3  )  =  P  (  x  ). Hence      Q    2    (  x  +  3  )  −  1  =  4  Q  (      x    2    −  4  x  +  4  )  =  4  Q  (  (  x  −  2      )    2    )Let   x  →  x  +  2      Q    2    (  x  +  5  )  =  4  Q  (      x    2    )  +  1So   Q  (  5  +  x  )  =  ±  Q  (  5  −  x  )Let   R  (  x  )  =  Q  (  5  +  x  ) . R is either odd or even and      R    2    (  x  )  =  4  R  (      x    2    −  5  )  +  1If R is odd then   R  (  0  )  =  0 and   R  (  5  )  =      1    4   and   R  (  −  20  )  =      15    64   and more generally :      S    0    =  0,       S          n      +      1        =  5  −      S    n    2  ,       T    0    =  0 and       T          n      +      1        =            1      −              T        n        2              4  It&#039;s easy to see that   P  (      S    n    )  =      T    n  , but   lim  (      T    n    ) is finite and   lim  (      |        S    n        |    )  =  ∞ is not a polynomial function.If R is even,   R  (  0  ) is an optimum (either maximum or minimum,       R    ′    (  0  )  =  0). Hence   R  (      5    ) is an optimum too, and       S    0    =  0,       S          n      +      1        =      5    +          S      n       , then       S    n   are all optimum... But all       S    n   are different, so       R    ′   has an infinite number of roots. This is not a polynomial function, except for a degree 0 polynomial.

Find all polynomials P(x) with real coefficents satisfying P^2(x)−1=4P(x^2−4x+1).

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