IIT JEE , AIEEE Forum - Mathematics , Algebra

man111

(2)

x³-x²-x-1 has a root a,b,c than prove that

a¹⁹⁹²-b¹⁹⁹²/a-b + b¹⁹⁹²-c¹⁹⁹²/b-c + c¹⁹⁹²-a¹⁹⁹²/c-a is an integer.

(3) solve _{[j=1 ]}^{[ 3]}x_j/a_i-a_j=1 (where i=1,2,3)

bhuwanaroracorroded

2

find a b c

sboosy

a^2-b^2 (a-b)(a+b)is divisible by a-b

a^3-b^3 (a-b)(a^2+ab+b^2)is divisible by a-b

similarly a^4-b^4 is also divisible by a-b

it can be thus shown that a^n-b^n is always divisble by a-b for natural number n

thus a^1992-b^1992/a-b is an integer

similarly the other two

so int+int+int which is also an integer

feynmann

Shoosy is wong !!!!

a, b, c arenot integers !! ( Two of them are infact complex )

hsbhatt

i think a, b and c are all real.

sboosy

I didnt read the first line my friends

so its my miss

anyway as far as i know mr feynmann is right

f(1)<0

f(2)>0

and by checking up we find only one root (real)

so i think there is only 1 real root

pls correct if wrong

konichiwa2x

This doesn't look good at all to me. The cubic equation

has only one real root, namely

. It has a pair of complex roots

. Notice that

is large (roughly

), while

and

are small (roughly

).

In the expression

, the two terms in

will tend to swamp the others. But

is real, and

has a substantial non-real component, since

. But i dont see how the expression could possibly be a real integer.