A-polynomial-function-f-x-satisfies-f-x-f-1-x-2f-x-2f-1-x-x-0-and-f-2-18-then-f-3-is-equal-to-

Question Number 21940 by Tinkutara last updated on 07/Oct/17

A polynomial function f (x) satisfies

f (x) f (\frac{1}{x}) = 2 f (x) + 2 f (\frac{1}{x}); x \neq 0 and

f (2) = 18, then f (3) is equal to

Commented by ajfour last updated on 07/Nov/18

N o o n e s o l v e d t h i s, i f c o r r e c t

i t s a g o o d o n e!

Answered by rahul 19 last updated on 08/Nov/18

56 ?

G o t t h e f u n c t i o n 2 (x^{3} + 1) \dots . .

Commented by ajfour last updated on 08/Nov/18

s t e p s p l e a s e, d o n t k n o w a n s w e r!

Answered by rahul 19 last updated on 08/Nov/18

L e t f (x) = a_{0} x^{n} + a_{1} x^{n - 1} + \dots . + a_{n - 1} x + a_{n} .

A c c . t o g i v e n c o n d i t i o n,

(a_{0} x^{n} + a_{1} x^{n - 1} \dots \dots + a_{n}) (\frac{a_{0}}{x^{n}} + \frac{a_{1}}{x^{n - 1}} + \dots \dots a_{n})

= 2 [(a_{0} x^{n} + a_{1} x^{n - 1} \dots . + a_{n}) + (\frac{a_{0}}{x^{n}} + \frac{a_{1}}{x^{n - 1}} \dots . + a_{n})]

\Rightarrow O n c o m p a r i n g c o e f f . o f x^{n},

a_{0} a_{n} = 2 a_{0} \Rightarrow a_{n} = 2

O n c o m p a r i n g c o e f f . o f x^{n - 1},

a_{n - 1} = 0 \dots . .

\dots \dots . . s i m i l a r l y, a_{n - 2} = \dots . . = a_{1} = 0

a n d a_{0} = \mp 2 .

∴ f (x) = 2 (\mp x^{n} + 1) .

N o w i n t h i s c a s e

f (2) = 18

\Rightarrow f (x) = 2 (x^{3} + 1)

\Rightarrow f (3) = 56 .

Commented by ajfour last updated on 08/Nov/18

Y e s R a h u l S i r, T h a n k s .

Answered by ajfour last updated on 08/Nov/18

(a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x + a_{0})

\times \frac{1}{x^{n}} (a_{n} + a_{n - 1} x + \dots . + a_{1} x^{n - 1} + a_{0} x^{n})

= \frac{1}{x^{n}} [a_{n} a_{0} x^{2 n} + (a_{n - 1} a_{0} + a_{n} a_{1}) x^{2 n - 1} + \dots .

\dots . . + (a_{n} a_{n} + a_{n - 1} a_{n - 1} + \dots + a_{0} a_{0} +) x^{n} + \dots

Missing or unrecognized delimiter for \left

= \frac{2}{x^{n}} [(a_{n} x^{2 n} + a_{n - 1} x^{2 n - 1} + \dots .

\dots . + a_{0} x^{n}) + (a_{0} x^{n} + a_{1} x^{n - 1} + \dots

\dots . + a_{n - 1} x + a_{n})]

c o m p a r i n g c o e f f s .

a_{n} a_{0} = 2 a_{n} \Rightarrow a_{0} = 2

a_{n - 1} a_{0} + a_{n} a_{1} = 2 a_{n - 1} \Rightarrow a_{1} = 0

a_{n} a_{2} + a_{n - 1} a_{1} + a_{n - 2} a_{0} = 2 a_{n - 2}

\Rightarrow a_{2} = 0

a n d s o o n . .

S o f (x) = a_{n} x^{n} + 2

f (1) \times f (1) = 4 f (1)

\Rightarrow f (1) = 0, 4

\Rightarrow a_{n} + 2 = 0, 4

\Rightarrow a_{n} = \pm 2

\Rightarrow f (2) = \pm 2 (2^{n} + 1) = 18

S o a_{n} = 2 & n = 3

f (3) = 2 (3^{3} + 1) = 56 .

Facebook Tweet Pin