Prove Theorems 17.9 and 17.10. Euclidean Algorithm for Polynomials let fix), g(x) F[x] with degree f(x) Then rk(x), the last nonzero remainder, is a greatest

Prove Theorems 17.9 and 17.10.
Euclidean Algorithm for Polynomials let fix), g(x) ˆˆ F[x] with degree f(x)

Then rk(x), the last nonzero remainder, is a greatest common divisor of f(x), g(x), and is a constant multiple of the monic greatest common divisor of f(x), g(x). [Multiplying rk(x) by the inverse of its leading coefficient allows us to obtain the unique monic polynomial we call the greatest common divisor.]
Let s(x) ˆˆ F(x), s(x) 0. Define relation R on F[x] by f(x) R g(x) if f(x) - g(x) = t(x)s(x), for some t(x) ˆˆ F[x] - that is, six) divides f(x) - g(x). Then R is an equivalence relation on F[x].

g(x) = q(x)f(x) + r(x), f(x) = qi (x)r(x) + ri (x), r(x) = q2(x)rl (x) + r2(x), degree r(x) degree f (x) degree ri(x)