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


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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

	$56 ?$ $G o t t h e f u n c t i o n 2 (x^{3} + 1) . . . . .$
	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

	$L e t f (x) = a_{0} x^{n} + a_{1} x^{n - 1} + . . . . + a_{n - 1} x + a_{n} .$ $A c c . t o g i v e n c o n d i t i o n,$ $(a_{0} x^{n} + a_{1} x^{n - 1} . . . . . . + a_{n}) (\frac{a_{0}}{x^{n}} + \frac{a_{1}}{x^{n - 1}} + . . . . . . a_{n})$ $= 2 [(a_{0} x^{n} + a_{1} x^{n - 1} . . . . + a_{n}) + (\frac{a_{0}}{x^{n}} + \frac{a_{1}}{x^{n - 1}} . . . . + a_{n})]$ $\Rightarrow O n c o m p a r i n g c o e f f . o f x^{n},$ $a_{0} a_{n} = 2 a_{0} \Rightarrow a_{n} = 2$ $O n c o m p a r i n g c o e f f . o f x^{n - 1},$ $a_{n - 1} = 0 . . . . .$ $. . . . . . . . s i m i l a r l y, a_{n - 2} = . . . . . = a_{1} = 0$ $a n d a_{0} = \mp 2 .$ $∴ f (x) = 2 (\mp x^{n} + 1) .$ $N o w i n t h i s c a s e$ $f (2) = 18$ $\Rightarrow f (x) = 2 (x^{3} + 1)$ $\Rightarrow f (3) = 56 .$
	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

	$(a_{n} x^{n} + a_{n - 1} x^{n - 1} + . . . + a_{1} x + a_{0})$ $\times \frac{1}{x^{n}} (a_{n} + a_{n - 1} x + . . . . + a_{1} x^{n - 1} + a_{0} x^{n})$ $= \frac{1}{x^{n}} [a_{n} a_{0} x^{2 n} + (a_{n - 1} a_{0} + a_{n} a_{1}) x^{2 n - 1} + . . . .$ $. . . . . + (a_{n} a_{n} + a_{n - 1} a_{n - 1} + . . . + a_{0} a_{0} +) x^{n} + . . .$ $. . . + (a_{1} a_{n} + a_{0} a_{n - 1}) x + a_{0} a_{n}]$ $= \frac{2}{x^{n}} [(a_{n} x^{2 n} + a_{n - 1} x^{2 n - 1} + . . . .$ $. . . . + a_{0} x^{n}) + (a_{0} x^{n} + a_{1} x^{n - 1} + . . .$ $. . . . + a_{n - 1} x + a_{n})]$ $c o m p a r i n g c o e f f s .$ $a_{n} a_{0} = 2 a_{n} \Rightarrow a_{0} = 2$ $a_{n - 1} a_{0} + a_{n} a_{1} = 2 a_{n - 1} \Rightarrow a_{1} = 0$ $a_{n} a_{2} + a_{n - 1} a_{1} + a_{n - 2} a_{0} = 2 a_{n - 2}$ $\Rightarrow a_{2} = 0$ $a n d s o o n . .$ $S o f (x) = a_{n} x^{n} + 2$ $f (1) \times f (1) = 4 f (1)$ $\Rightarrow f (1) = 0, 4$ $\Rightarrow a_{n} + 2 = 0, 4$ $\Rightarrow a_{n} = \pm 2$ $\Rightarrow f (2) = \pm 2 (2^{n} + 1) = 18$ $S o a_{n} = 2 & n = 3$ $f (3) = 2 (3^{3} + 1) = 56 .$
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