In this exercise, we will prove the Division Algorithm for polynomials. Let R[x] be the ring...

In this exercise, we will prove the Division Algorithm for polynomials. Let R[x] be the ring of polynomials with real coefficients. For the purposes of this exercise, extend the definition of degree by deg(0) = −1. The statement to be proved is: Let f(x),g(x) ∈ R[x][x] be polynomials with g(x) ? 0. Then there exist unique polynomials q(x) and r(x) such that

f (x) = g(x)q(x) + r(x) and deg(r(x)) < deg(g(x)).

Fix general f (x) and g(x).

(a) Let S = { f (x) − g(x)s(x) | s(x) ∈ R[x][x]}. Prove that if h1(x) ∈ S and deg(h1(x)) ≥ deg(g(x)), then there is an

h2(x) ∈ S with deg(h2(x)) < deg(h1(x)).
(b) Show: If h1(x), h2(x) ∈ S with deg(h1(x)) = deg(h2(x)), then there is an h3(x) ∈ S with deg(h3(x)) < deg(h1(x)).
(c) Prove S has a unique element of minimal degree.
(d) Verify the existence of q(x) and r(x).

Solutions

Expert Solution

I have given the proof by a different method. Hope you will love it and then take your time and try to solve on your own.

Have a good day!!!

Colby Messinger answered 1 year ago

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