Another Math Q.?

Question

f(x) = x^3 - 4(x^2) - 16x +60

AND: its given that f(x) = y

f(y) = z

and f(z) = x
Find all x, y , z belonging to integers.

Isnt there a simple solution based on number theory.

Answer 1

First you can factor f(x)=(x-4)(x-4)(x+4)-4. So (x-4)(x-4)(x+4)=(y+4). If you write out all the equations you'll see that to get back to x, either (x+4)=0 or (x-4) has to be +1 or -1. So the positive solutions are x=3 and x=5. There's also the negative solution x=-4 since then trivially f(x)=x.

Answer 2

we can substiute y for f(x) and f(z) for x to get
y = [f(z)]^3- 4[f(z)]^2-16 f(z) + 60
We can evaluate f(x) for a given value of x.
This gives us a value of y, since f(x)=y
We can solve the above equation for f(z) , which should give 3 values of z (hopefully)