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

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Advance Math

In this exercise, we will prove the Division Algorithm forpolynomials. Let R[x] be the ring of polynomials with realcoefficients. For the purposes of this exercise, extend thedefinition 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 thereexist 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).

  1. (a) Let S = { f (x) ? g(x)s(x) | s(x) ? R[x][x]}. Prove that ifh1(x) ? S and deg(h1(x)) ? deg(g(x)), then there is an

    h2(x) ? S with deg(h2(x)) < deg(h1(x)).

  2. (b) Show: If h1(x), h2(x) ? S with deg(h1(x)) = deg(h2(x)), thenthere is an h3(x) ? S with deg(h3(x)) < deg(h1(x)).

  3. (c) Prove S has a unique element of minimal degree.

  4. (d) Verify the existence of q(x) and r(x).

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