Selecting any other pair of quadratic conjugate roots
, we see that they should be -b [+ or -] e' [square root of D] with rational b and e' > 0.
In this section, only the case when system has a real root and conjugate roots
will be analyzed.
Let q be the companion polynomial of p, and let it have at least one pair of complex conjugate roots c = u [+ or -] vi, where u, v [member of] R, v > 0.
Since there are at most [n.sub.1] [less than or equal to] n pairs of complex conjugate roots, p may have at most [n.sub.1] zeros derived from complex zeros of q.
* one real root and a pair of complex conjugate roots.
There is one real root apparent at x = 1 but no indication whatsoever of the whereabouts of the complex conjugate roots. In this particular instance it is easy enough to divide the expression for y by the known factor (x - 1) to yield y = (x - 1)([x.sup.2] + 4x + 5) from which the remaining two roots are found to be -2 [+ or -] i.
Note that the root of BCH codes is consecutive, so for the codes with the error-correcting capacity t, all the LLRs [phi]([H.sub.i]) corresponding to the roots [[alpha].sup.i] (i [less than or equal to] 2t - 1) should be lower than the others except the conjugate roots
of every [[alpha].sup.i] (i [less than or equal to] 2t - 1).
The points of intersection are (-2, 1) and (-2, -1), so the complex conjugate roots
are z = -2 [+ or -] i.
Case 3: e = 1 and [beta] < 1 (complex conjugate roots
This also leads to the n-th roots of unity, although the location of these n-roots relative to the Cartesian plane is commonly misunderstood (see Stroud (1986, Programme 2, Theory of Equations); he (incorrectly) places the complex conjugate roots
in the Cartesian plane at one of the turning points of a cubic equation.) In fact, Equations (1) and (2) are essentially the same, since the possibility of complex solutions should most definitely be entertained for Equation (2).
which is recognised as the total distance along the H'-axis between a pair of conjugate roots
two-dimensional x-y plot of the quadratic equation does not reveal the location of the complex conjugate roots
, and the interested student might well be forgiven for asking, "Where exactly are the roots located and why can't I see them?" In the author's experience, this sort of question is hardly ever raised--or answered satisfactorily--in school Years 11 or 12, or in undergraduate mathematics courses.