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If f(x)f(1÷x)=f(x)+f(1÷x).Then Why f(x)=1+x^n?Here f(x) is a polynomial function.

If f(x)f(1÷x)=f(x)+f(1÷x).Then Why f(x)=1+x^n?Here f(x) is a polynomial function.

Grade:12

1 Answers

mycroft holmes
272 Points
4 years ago
Multiply the equation by xn to obtain.
x^n f(x) f \left(\frac{1}{x} \right) = x^n f(x)+ x^n f \left(\frac{1}{x} \right )
 
Notice that if is a root of f(x), it is also a root of the ‘reciprocal polynomial’ xn f(1/x) and vice-versa. This means that
 
f(x) = c x^n f \left(\frac{1}{x} \right) 
 
for some complex number c.
 
Using this above relation in the original equation gives
f^2(x) = f(x) (1+cx^n)
 
So either f(x) is the zero polynomial or f(x)= (1+cx^n)
 
Again plugging back in the original equation and solving for c, we get c2 =1 so c=1, or -1.
 
So that the only solutions in polynomials is f(x)= 1 \pm x^n

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