D dhiraj New member Joined Jul 18, 2014 Messages 5 Jul 18, 2014 #1 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.
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.
D dhiraj New member Joined Jul 18, 2014 Messages 5 Jul 18, 2014 #2 Denis said: 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 Click to expand... 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?
Denis said: 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 Click to expand... 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?
