let R[x] have the ordering given by 

i. P_low     

ii. P_High 

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. x - 3x² + 5x³ 

b. 2 - 3x² + 5x³ 

c. x - 3x² + 4x³ 

d. x + 3x² + 4x⁴ 

e. x + 3x² - 4x³ 

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, P_low and P_high, of positive elements. A nonzero polynomial in R[x] can be written in the form f(x) = a_rX^r + a_r+1x^r+1 + · · · + a_nXⁿ where ar ≠ 0 and a_n ≠ 0, so that a_rx^r and a_nxⁿ are the nonzero terms of lowest and highest degree, respectively. Let P_low be the set of all such f(x) for which a_r ∈ P, and let P_high be the set of all such f (x) for which a_n ∈ P. The closure and trichotomy requirements that P_low and P_high 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 P_low, the polynomial f(x) = -2x + 3x⁴ would not be positive because -2 is not positive in Z. With ordering given by P_high, this same polynomial would be positive because 3 is positive in Z.