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Find all real numbers $x$ such that $f(x) = f(f(x))$, where $f(x) = x^2 - 3x + x^3 - 7x^2 + 5x.$

 Jul 26, 2024
 #1
avatar+129842 
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f(x) =  x^3 -6x^2 + 2x 

 

f (f(x)) =  [ x^3 - 6x^2 + 2x]^3 - 6 [ x^3 -6x^2 + 2x]^2 + 2[ x^3 - 6x^2 + 2x] =

 

x^9 - 18 x^8 + 114 x^7 - 294 x^6 + 300 x^5 - 312 x^4 + 154 x^3 - 36 x^2 + 4 x

 

So

 

x^3 -6x^2 + 2x  = x^9 - 18 x^8 + 114 x^7 - 294 x^6 + 300 x^5 - 312 x^4 + 154 x^3 - 36 x^2 + 4 x

 

x^9 - 18 x^8 + 114 x^7 - 294 x^6 + 300 x^5 - 312 x^4 + 153 x^3 - 30 x^2 + 2 x =  0

 

With a little help from WolframAlpha :

 

Solutions are :

 

 

 

cool cool cool

 Jul 26, 2024

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