Question

please provide a proof and explanation for

The division algorithm for F[x]. Just the existence part only, not the uniqueness part.

Answer 1

Let f(x)= anx^ n + an−1x^ n−1 + · · · + a1x + a and g(x)= bmx^ m + bm−1x^ m−1 + · · · + b1x + b0 are polynomial of degree n and m respectively.

We proceed by induction on the degree n of f(x).

If the degree n of f(x) is less than the degree m of g(x), there is nothing to prove, take q(x) = 0 and r(x) = f(x).

Suppose the result holds for all degrees less than the degree n of f(x). Put q0(x) = cx^(n−m), where c = an/bm. Let f1(x) = f(x) − q1(x)g(x). Then f1(x) has degree less than g. By induction then, f1(x) = q1(x)g(x) + r(x), where r(x) has degree less than g(x). It follows that f(x) = f1(x) + q0(x)g(x) = (q0(x) + q1(x))g(x) + r(x) = q(x)g(x) + r(x).

Hence by induction , division algorithm holds for f(x)