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Let $$f(x)=x^2-3$$ For how many integer values of x is $f(f(f(x)))$ divisible by x?

Thomas Andrews
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king
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  • I don't know how to solve it, but I know what I would do to try to find a solution. The first thing would be to insert $x=0,1,2,3$ or $4$ and see what pops out. You should try that. And please let us know the results. – Arthur Jan 30 '19 at 16:20
  • In general, if $h$ is an integer polynomial and $x$ is an integer, then $x$ is a divisor of $h(x)-h(0).$ So $x$ is a divisor of $h(x)$ if and only if $x$ is a divisor of $h(0).$ – Thomas Andrews Jan 30 '19 at 17:15
  • Could you give me more hints? do you mean I should solve for h(0)? – king Jan 31 '19 at 01:13

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Hint: $$f(f(x))=(x^2-3)^2-3$$ and now what is $$f(f(f(x))$$?

Dr. Sonnhard Graubner
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  • The f(f(f(x)) becomes very complicated, I don't know the next step to solve this problem – king Jan 31 '19 at 01:06