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Polynomial Division and Greatest Common Divisors
| Content Provider | Semantic Scholar |
|---|---|
| Copyright Year | 2008 |
| Abstract | 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 |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://web.cs.iastate.edu/~cs577/handouts/polydivide.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
