Polynomials ------------------

Let ζ be the set of all polynomials with real coefficients.
Let f,g ∈ ζ \{0} and
A = { mf + ng: m,n ∈ ζ }
Suppose r ∈ A \ {0} has the property that
deg r =< deg p for all p ∈ A \ {0}

a) Show that r divides every polynomial in A
Deduce that r is a G.C.D. of f and g (i.e. r divides both f and g, and if h divides
both f and g then h divides r)

b) Let B= {hr: h ∈ ζ }
Show that A=B

c) If deg r=0, i.e. r is non-zero constant, show that there exist m_0 , n_0 ∈ ζ
such that (m_0)(f) +(n_0)(g) = 1
and also A = ζ