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20 1. Fundamentals

7. 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.)

8. Why is the general method of solving a cubic equation not a part of
   most school curricula?


9. 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.

10. 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.

11. 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.