greatest common divisor

[dhiraj]

find 'c' if gcd of x^3+cx^2-x+2c & x^2+cx-2 is a linear polynomial?

Please help as I am not able to solve it on my own.

 

[dhiraj]

Hint:
(x^3 + cx^2 - x + 2c) / (x^2 + cx - 2)

= x + x^(-1), remainder c + 2x^(-1)
= x + 1/x, remainder c + 2/x

Sir couldn't understand the philosophy clearly. will you please elaborate some more?

however what i did is presented below:

let f(x) = x^3 + cx^2 - x + 2c and g(x) = x^2 + cx - 2.
further let h(x) = ax+b is the linear polynomial which is the gcd of f(x) and g(x).

now since h(x) divides g(x), so h(x) will also divide x*g(x).

h(x) will also divide f(x)-g(x) [using gcd of two numbers also divides their difference].

so h(x) will also divide f(x)-x*g(x) i.e. h(x) will divide x+2c.

now what? please suggest. is there any flaw in the methdology adopted here?