Thread: Division Theorem for Polynomials...

Prove the Following Lemma...:

Suppose p(x) and d(x) are polynomials in F[x] and that d(x) has degree at least 1. There exist unique polynomials q(x), r(x) E F(x) such that r(x) has lower degree than d(x) and
p(x) = q(x) . d(x) + r(x)



Please help..I have been struggling to understand this Lemma, but its quite complicated.

Hello,

Try a proof by contradiction for it...
This is similar to the Euclidian algorithm.

Originally Posted by vikramtiwari

Prove the Following Lemma...:

Suppose p(x) and d(x) are polynomials in F[x] and that d(x) has degree at least 1. There exist unique polynomials q(x), r(x) E F(x) such that r(x) has lower degree than d(x) and
p(x) = q(x) . d(x) + r(x)
.

If d(x) (non-zero) has degree zero then it is a constant polynomial, and there is nothing to show, so it is safe to assume it is a linear polynomial. Consider the set S={ p(x) - l(x).d(x)| l(x) in F[x]}. By well-ordering it means there is a least degree for this set, thus, there exists q(x) so that p(x) - q(x).d(x) = r(x) is smallest degree. Argue that r(x) is either zero polynomial or else deg r(x) < 1.