The problem involves a polynomial function f(x) that satisfies the equation f(x)·f(1/x) = f(x) + f(1/x) with the condition that f(3) = 28. The goal is to determine the value of f(4).
The discussion is ongoing, with participants providing hints and guidance on how to approach the problem without giving direct solutions. There is recognition of the need to explore the polynomial's coefficients and degree further.
Participants note that the rules of the forum prohibit providing complete solutions, emphasizing the importance of individual effort in solving the problem. There is also a mention of potential misinterpretation of the notation used in the problem statement.
mr newtein said:Homework Statement
f(x) is a polynomial satisfying
f(x).f(1/x)=f(x)+f(1/x),f(3)=28 than f(4) is?
Homework Equations
The Attempt at a Solution
okay i have assumed a polynomial ,f(3) is given,so can i calculate coefficeints and degree of polynomial, and calculate f(4)
How does what you wrote apply to this problem? The OP wrote "f(x).f(1/x)=f(x)+f(1/x)", not f(1/x) = f(x) + f(1/x).Contingency said:f(1/x)=f(x)+f(1/x) => f(x)=0 for all x≠0
are you sure you copied correctly?
than what's the solution and question is correct.that dot is multiplicationHallsofIvy said:Yes, but Contingency interpreted the "." as a period.
mr newtein said:than what's the solution and question is correct.that dot is multiplication
thanks bro i got the hint,and at last solutionInfinitum said:By the PF rules, we cannot give you the solution. You have to do it yourself.
For starters, as Mark44 said, you cannot know the degree of the polynomial. So let it be a polynomial of degree n, with general coefficients. Now try finding the value of the function for a particular x, that will give you the sum of coefficients.
Using the relation f(x)\cdot f(1/x) = f(x) + f(1/x) for your assumed polynomial, try to deduce what the function can be by comparing coefficients.