The discussion revolves around finding the value of f(f(2)) for a polynomial function f(x) that satisfies the equation 2 + f(x)f(y) = f(x) + f(y) + f(xy), given that f(2) = 5. Participants utilized substitutions and polynomial forms, ultimately deducing that f(x) can be expressed in terms of g(x) with roots leading to the conclusion that g(x) = ax^k, where a = 1. The degree of the polynomial and the relationships between coefficients were also explored, culminating in a deeper understanding of polynomial behavior.
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f(x)=g(x)+1haruspex said:Exactly. So, what is the general form of g(x), and thus, the general form of f(x)?
Well, yes, that's the relationship between f(x) and g(x), but what is the general form of g(x)? We have shown that all its roots are 0, right? So what does that polynomial look like?utkarshakash said:f(x)=g(x)+1
haruspex said:Well, yes, that's the relationship between f(x) and g(x), but what is the general form of g(x)? We have shown that all its roots are 0, right? So what does that polynomial look like?
Indeed I did, but from your post #29 it seemed like you'd not been reading all those exchanges, perhaps because you wanted to figure it for yourself with a few hints.utkarshakash said:You have already stated that in an earlier post
haruspex said:Indeed I did, but from your post #29 it seemed like you'd not been reading all those exchanges, perhaps because you wanted to figure it for yourself with a few hints.
So, do you understand why g(x) = axk for some a and k? Do you understand how to determine a from the equation for g(), and then the value of k from the given datapoints?
utkarshakash said:I do not know how to derive a but assuming a=1, I can find k.
In your post 27 you wrote, correctly,utkarshakash said:I do not know how to derive a but assuming a=1, I can find k.
Substitute g(x) = axk in there.g(x)g(y)=g(xy)
haruspex said:In your post 27 you wrote, correctly,
Substitute g(x) = axk in there.