François Viète

Exercises

1. Viète states, in Proposition XI on powers of a binomial, that (A + B)³ = A³ + 3AB² + 3A²B+ B³, with similar expansions for the fourth, fifth and sixth powers.  Give detailed demonstrations of these expansions.
2. 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 - r)  as follows:

(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.
3. 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.
4. Here is problem v.7 in Diophantus' Arithmetica: We 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 - E).