Polynomial Division and Greatest Common Divisors
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چکیده
It is easy to see that there is at most one pair of polynomials (q(x), r(x)) satisfying (1); for if (q1(x), r1(x)) and (q2(x), r2(x)) both satisfy the relation with respect to the same polynomial u(x) and v(x), then q1(x)v(x)+r1(x) = q2(x)v(x)+r2(x), so (q1(x)− q2(x))v(x) = r2(x)−r1(x). Now if q1(x)− q2(x) is nonzero, we have deg((q1 − q2) · v) = deg(q1 − q2)+deg(v) ≥ deg(v) > deg(r2 − r1), a contradiction; hence q1(x)− q2(x) = 0 and r1(x) = r2(x). Given its uniqueness, we denote q(x) = ⌊ v(x) ⌋, analogous to the quotient in integer division. Obviously, r(x) = u(x)− v(x)⌊ v(x) ⌋. Let
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