Let f(x) be a polynomial such that f(f(x))−x^2 =xf(x). Find f(-100)

Answer 1

If the degree of the polynomial f(x) is d, then the degree of f(f(x)) is d^2. If d > 1, then f(f(x)) - x^2 has degree d^2, while x f(x) has degree d+1. But d^2 = d+1 has no integer solutions. Therefore d must be equal to 1, this means that:

f(x) = a x + b

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

a (a x + b) + b -x^2 =

a^2 x + a b + b -x^2

This has to be equal to

x f(x) = a x^2 + b x

for all x, therefore:

a = -1

b = a^2 = 1

a b + b must be equal to zero, and this indeed the case.

So, f(x) = -x + 1.

Answer 2

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