MY CART (5)

Use Coupon: CART20 and get 20% off on all online Study Material

ITEM
DETAILS
MRP
DISCOUNT
FINAL PRICE
Total Price: Rs.

There are no items in this cart.
Continue Shopping
Menu
Grade: 12
        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.
3 years ago

Answers : (1)

mycroft holmes
272 Points
							
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
3 years ago
Think You Can Provide A Better Answer ?
Answer & Earn Cool Goodies


Course Features

  • 731 Video Lectures
  • Revision Notes
  • Previous Year Papers
  • Mind Map
  • Study Planner
  • NCERT Solutions
  • Discussion Forum
  • Test paper with Video Solution


Course Features

  • 101 Video Lectures
  • Revision Notes
  • Test paper with Video Solution
  • Mind Map
  • Study Planner
  • NCERT Solutions
  • Discussion Forum
  • Previous Year Exam Questions


Ask Experts

Have any Question? Ask Experts

Post Question

 
 
Answer ‘n’ Earn
Attractive Gift
Vouchers
To Win!!! Click Here for details