division algorithm

Let us learn about division algorithm

In mathematics, polynomials in more than one variable do not form a Euclidean domain, so it is not possible to construct a true division algorithm. Given a polynomial g, polynomials (f₁, ..., f_m) and an order on the monomials in k[x₁, ..., x_n], we construct the reduction of g modulo f₁, ..., f_m by the following algorithm. Let a_i denote the leading term of f_i with respect to the monomial order. Repeatedly apply the following until no monomial term of g is divisible by any of the a_i :

Division Algorithm for Polynomials

Let F is a field, f(x), g(x) e F[x] with g(x) ≠ 0.

In q(x), r(x) in the F[x] such that f(x) = g(x) q(x) + r(x) and r(x) =0

Proof :-
(1) Let f(x) = g(x) q(x) + r(x) with g(x) ≠ 0

(2) If f(x) = 0 or deg f(x) is less than g(x), then r(x)=f(x), q(x)=0.

(3) So, we assume that f(x) ≠ 0 and deg f(x) ≥ deg g(x)

(4) Let f(x) = a_nxⁿ + ... + a₀

(5) Let g(x) = b_mx^m + ... + b₀

(6) Let f₁(x) = f(x) - a_nb_m^-1x^n-mg(x)

(7) We note that deg f₁(x) is <>deg f(x) since:

f(x) - a_nb_m^-1x^n-mg(x)

= a_nxⁿ + ... + a₀ - a_nb_m^-1x^n-m(b_mx^m + ... + b₀)

= a_nⁿ - a_nxⁿ + a_n-1x^n-1 + ... + a₀ - a_nb_m^-1b_m-1x^n-1 - ... - a_nb_m^-1b₀x^n-m

= a_n-1x^n-1 + ... + a₀ - a_nb_m^-1b_m-1x^n-1 - ... - a_nb_m^-1b₀x^n-m

So that f₁(x) has a degree of n-1 while f(x) has a degree of n.

In our next blog we shall learn about cleavage furrow I hope the above explanation was useful.Keep reading and leave your