91Ó°ÊÓ

To solve the fourth-degree equation \(x^{4}-p x^{2}-q x-\) \(r=0\), Descartes considered the cubic equation in \(y^{2}\). \(y^{6}-2 p y^{4}+\left(p^{2}+4 r\right) y^{2}-q^{2}=0 .\) If \(y\) is a solution, show that the original polynomial factors into two quadratics: \(r_{1}(x)=x^{2}-y x+\frac{1}{2} y^{2}-\frac{1}{2} p-\frac{q}{2 y}, r_{2}(x)=x^{2}+y x+\) \(\frac{1}{2} y^{2}-\frac{1}{2} p+\frac{q}{2 y}\), each of which can be solved. Apply this method to solve the equation \(x^{4}-17 x^{2}-20 x-6=0\) Note that the corresponding equation in \(y, y^{6}-34 y^{2}+\) \(313 y^{2}-400=0\), has the solution \(y^{2}=16\)

Pascal stated that the odds in favor of throwing a six in four throws of a single die are 671 to 625 . Show why this is true.

Show that the odds against at least one 1 appearing in a throw of three dice is \(125: 91\). (This answer was stated by Cardano.)

For a roll of three dice, show that both a 9 and a 10 can be achieved in six different ways. Nevertheless, show that the probability of rolling a 10 is higher than that of rolling a \(9 .\) (A discussion of this idea is found in a fragment of a work of Galileo.)

If two players play a game with two dice with the condition that the first player wins if the sum thrown is 7 , the second wins if the sum is 6 , and the stakes are split if there is any other sum, find the expectation (value of the chance) of each player.

Outline a lesson on the principle of mathematical induction using material from Pascal's Treatise on the Arithmetical Triangle.