Question

Prove: Suppose that f(x)=g(x)q(x)+r(x). Then d(x) is a gcd of f(x) and g(x) iff d(x) is the gcd of g(x) and r(x).

Answer 1

let gcd of [g(x) and r(x)]=z(x)
so g(x)=z(x)l(x)
r(x)=z(x)k(x)
f(x)=g(x)q(x)+r(x)
f(x)=z(x)[l(x)q(x)+r(x)
so gcd of f(x) and g(x) is z(x)