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

vagnhestagn 2022-10-08 Answered
Find all polynomials P ( x ) with real coefficents satisfying P 2 ( x ) 1 = 4 P ( x 2 4 x + 1 )
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Answers (1)

recepiamsb
Answered 2022-10-09 Author has 9 answers
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 ) + 1
So 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 ) + 1
If 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'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.
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