20 1. Fundamentals
- On August 7, 1877 (?), Arthur Cayley (1821-1895) wrote to Rudolf
Lipschitz (1832-1903) a letter containing the following paragraph:
As to the cubic, there is a variation of Cardano’s solu-
tion which I think is theoretically interesting: if instead of
assuming x = a + b, we assume x = a2b + ab2, then instead
of ab and a3 + b3, we have only a3b3 and a3 + b3 rationally
determined: a and b may therefore be any values whatever
of the cube roots of a3, b3, but the apparently g-valued
function a2b + ab2 will be only 3-valued.
(Rudolf Lipschitz, Briefwechsel mit Cantor, Dedekind,
Helmholtz, Kronecker, Weierstrass. DMV, Vieweg & Sohn,
1986)?
Verify that Cayley’s remark is true. (Cayley’s x, a, b correspond to
our t, u, v in Exercise 4.)
- Why is the general method of solving a cubic equation not a part of
most school curricula? - Find the relationship between p and q in order that the equation
x3 + px + q = 0 may be put into the form
x4 = (x2 + ox + b)2.
Hence, solve the equation
8x3 - 36x + 27 = 0.
- Vieta [Cajori, History, p. 1381 had an alternative method of solving
a cubic of the form
x3 - 3a2x = a2b
where a, b are real numbers which satisfy Ibl 5 2]a]. Show that, if 4
is defined by b = 2a cos 4, then a solution of the equation is given by
x = 2a cos(l/3)4. Use this method to locate a solution of each of the
following equations:
(a) x3 - 3a2x = 0.
(b) x3 - 3x - 2 = 0.
- The quartic equation: Descartes’ Method (1637).
(a) Argue that any quartic equation can be solved once one has a
method to handle quartic equations of the form
t4 +pt2 + qt + r = 0.
‘Used with permission of the editor, Dr. Winfried Scharlau, and the publisher.