A Nice and EZ Polynomial Equation

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❤️ P(P(x)+1)=9x+7

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[lazymello]

Another way of showing that P(x) is linear is assuming a is a root of P (maybe complex)
So P(a)=0
then P(P(a)+1)=9a+7
P(1)=9a+7
Now since P is a function, P(1) has a unique value, so 9a+7 has a unique value, so a has a unique value.
i.e, P has a unique root, hence it is linear. Notably, this can be used to prove uniqueness of the root even if P is not a polynomial, so it may help in generalising the functional equation

edit: one slight correction, for a general function the root may or may not exist. i.e no. of roots are either 0 or 1

[mathmode]

We can look at this equation as a quadratic for P(x) and get the same result

[bjorntorlarsson]

You leave Sybermath alone for a day, and when you come back he has invented these horrors of horrors! I suppose substitution will somehow do the trick.