Let f(x) be a polynomial such that
f(f(x))−x^2=xf(x).
Find f(−100).
101
How
To find the value of f(-100), we need to solve the given equation and substitute x = -100.
Let's first solve the given equation step-by-step:
f(f(x)) - x^2 = xf(x)
Substituting x = -100:
f(f(-100)) - (-100)^2 = -100f(-100)
Simplifying further:
f(f(-100)) - 10000 = -100f(-100)
Now, we need to make some observations and simplifications to find the value of f(-100).
Observation 1:
Note that f(-100) is an input to the function f. Let's assume f(-100) = a, where 'a' is some constant.
Observation 2:
Substituting f(-100) = a in the equation:
f(f(-100)) - 10000 = -100f(-100)
f(a) - 10000 = -100a
Simplifying further:
f(a) + 100a - 10000 = 0
Now, we have a polynomial equation f(a) + 100a - 10000 = 0, where 'a' is a constant.
To find the value of f(a), we need either more information about the polynomial or additional equations.
Unfortunately, with the given equation alone, we cannot determine the exact value of f(-100) or the polynomial f(x).