# Math Help - polynomial functions

1. ## polynomial functions

the number of polynomial functions $F$of degree $>=1$ satisfy $F(x^2 ) = [F(x)]^2 = F(F(x)) is$
(A) 0
(B) 1
(C) 2
(D) infinetly many

2. Originally Posted by grgrsanjay
the number of polynomial functions $F$of degree $>=1$ satisfy $F(x^2 ) = [F(x)]^2 = F(F(x)) is$
(A) 0
(B) 1
(C) 2
(D) infinetly many
I get answer (B).

f(x^2) = f(x)f(x) = f(f(x))

Consider just the part

f(x^2) = f(f(x))

So try x^2 = f(x).

Verify that this does not contradict the other equality given.

Given the constraint on the degree we cannot choose f(x) = 0 or f(x) = 1.

It also happens that 2 is the only nonzero real number satisfying n*n = n+n. (Look at the exponent to see why I say this.)

3. so if the degree of polynomial can me anything the f(x)=0 can also be an answer
$f(x) = x^2$ satisfy all three parts

this unfortunately appeared in my maths Olympiad conducted yesterday