Find f(2) of a Polynomial Function | R and f(2)=5

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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) and the condition f(2) = 5. Participants explore substitutions and relationships between coefficients of the polynomial, ultimately deducing that f(x) can be expressed in terms of another function g(x). The conversation highlights the implications of roots in polynomial functions and the constraints they impose, leading to the conclusion that g(x) must take a specific form. The final consensus is that understanding the roots and structure of g(x) is crucial for determining the polynomial's degree and coefficients.
  • #31
haruspex said:
Exactly. So, what is the general form of g(x), and thus, the general form of f(x)?
f(x)=g(x)+1
 
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  • #32
utkarshakash said:
f(x)=g(x)+1
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?
 
  • #33
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?

You have already stated that in an earlier post
 
  • #34
utkarshakash said:
You have already stated that in an earlier post
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?
 
  • #35
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?

I do not know how to derive a but assuming a=1, I can find k.
 
  • #36
utkarshakash said:
I do not know how to derive a but assuming a=1, I can find k.

You would able to derive it if you used some other properties of f(x) you figured out before.
 
  • #37
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,
g(x)g(y)=g(xy)
Substitute g(x) = axk in there.
 
  • #38
haruspex said:
In your post 27 you wrote, correctly,

Substitute g(x) = axk in there.

Thanks!

PS-This was the longest thread I have ever started.
 

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