let R[x] have the ordering given by i. P low ii. P High as described in Example

Question:

let R[x] have the ordering given by 

i. Plow     

ii. PHigh 

as described in Example 25.2. In each case (i) and (ii), list the labels a, b, c, d, e of the given polynomials in an order corresponding to increasing order of the polynomials as described by the relation < of Theorem 25.5.

a. 4x - 3x2 

b. 4x -t-2x

c. 4x - 6x3 

d. 5x - 6x

e. 3x - 2x2


Data from Example 25.2


Let R be an ordered ring with set P of positive elements. There are two natural ways to define an ordering of the polynomial ring R[x]. We describe two possible sets, Plow and Phigh, of positive elements. A nonzero polynomial in R[x] can be written in the form f(x) = arXr + ar+1xr+1 + · · · + anXn where ar ≠ 0 and an ≠ 0, so that arxr and anxn are the nonzero terms of lowest and highest degree, respectively. Let Plow be the set of all such f(x) for which ar ∈ P, and let Phigh be the set of all such f (x) for which an ∈ P. The closure and trichotomy requirements that Plow and Phigh must satisfy to give orderings of R[x] follow at once from those same properties for P and the definition of addition and multiplication in R[x]. Illustrating in Z[x], with ordering given by Plow, the polynomial f(x) = -2x + 3x4 would not be positive because -2 is not positive in Z. With ordering given by Phigh, this same polynomial would be positive because 3 is positive in Z.

Fantastic news! We've Found the answer you've been seeking!

Step by Step Answer:

Related Book For  book-img-for-question
Question Posted: