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Viète states, in Proposition
XI on powers of a binomial, that (A + B)3
= A3 + 3AB2 + 3A2B+
with similar expansions for the fourth, fifth and sixth powers. Give
detailed demonstrations of these expansions.
Use the result in the Corollary
to Theorem I at the end of the selected text to give a proof of the
geometric series formula a + ar + ar2
+ ar3 + ... = a/(1 -
(a) The algebraic formulation of
this result is given in the note at the text there (note
12). Put A = 1 and B = r in this formula;
what do you get? Multiply by a. (b) The geometric
series converges to a sum precisely when -1
< r < 1. As the largest power n in the formula
increases, what happens to the value of the power of r that appears
here? Let n increase to infinity; how then is this formula
affected? This will yield the geometric series formula.
Proposition XIII in The Analytic
Art (not in our selection) states: The square of the sum of two
roots minus the square of their difference equals four times the product
of the roots. Demonstrate this.
Here is problem v.7 in Diophantus'
wish to find two numbers such that their sum and the sum of their cubes
are equal to two given numbers. Follow Viète's solution. (a)
Let the given sum of the numbers be called G, the given sum of their
cubes be D, and the (unknown as yet) product of the numbers be called
A. If the unknown numbers are x and y, use the
binomial theorem to show that G3 = D + 3AG.
(b) Solve this last equation to find A, so that now A
is known. (c) Let E be the (unknown as yet) difference
of x and y. Use #3 above to solve for E.
(d) Finally, since we know G and E, we can solve for
the numbers x and y by using Proposition VI. Show then
that x and y are given by the quantities 1/2(A + E)
and 1/2(A -
last modified 10/16/00
Copyright (c) 2000. Daniel E. Otero