Re: Reference for a cubic with a double root?

From: Ken Pledger (Ken.Pledger_at_mcs.vuw.ac.nz)
Date: 08/11/04 

Date: Thu, 12 Aug 2004 08:34:31 +1200

In article <cfcf5h$9em$1@panix1.panix.com>,
 baloglou@panix.com (George Baloglou) wrote:

>
> .... I still need a reference where the condition
> is literally spelled out, *not for the special case b = 0*, but for the
> general case, *preferably even when a is not taken to be equal to 1* :-)
> [I see that this is done in (my Aristotle University professor) Konstantinos
> Lakkis's "Algebra" (1976), but that book is written in Greek, and I need a
> reference in English; C. C. MacDuffee's "Theory of Equations" (1954) comes
> close to what I need in Theorem 50* (p. 91), except that he assumes a = 1
> (a very minor offense which I would still prefer to avoid if possible).]
>
> *With D = (b^2)(c^2) - 4(c^3) - 4(b^3)d - 27(d^2) + 18bcd, the cubic
> x^3 + bx^2 + cx + d has a double (real) root in case D = 0, three distinct
> real roots in case D > 0, and precisely one real root in case D < 0 ....

      For such traditional algebra I usually look at S. Barnard & J. M.
Child, "Higher Algebra," Macmillan, 1936 and reprints. Their Chapter
XII begins (on pp.179-180) with the general cubic

a(x^3) + 3b(x^2) + 3cx + d = 0

and moves on quickly to the discriminant and its properties. This seems
exactly what you want, except for the two coefficients where they
introduce factors of 3 to simplify various formulae.

            Ken Pledger.