What Girard meant by the last statement about signs is that one must first arran

Question

What Girard meant by the last statement about signs is that one must first arrange the equation so that the degrees alternate on each side of the equation. Thus, x 4=4x3+7x2-34x-24 should be rewrite as x4-7x-24=4x2-4x3. Three roots of this equation being 1,2,-3, and 4, the first faction is equal to 4, the coefficient of x3; the second to -7, the coefficient of x2; the third to -34, the coefficient of x; and the fourth to -24, the constant term. In the first part of the theorem, Girard was asserting the truth of the fundamental theorem of algebra of algebra, that every polynomial equation has a number of solutions equal to its degree (denomination of the highest quantity). As this examples shown, he acknowledge that a given solution could occur with multiplicity greater than 1.He also fully realized that in his count of solution he would have to include imaginary ones (which he called "impossible"). So in his example x4+3 =4x, he noted that the four factions are 0,0,4,3. Because 1 is a solution of multiplicity 2, the two remaining solutions have the property that their product is 3 and their sum is -2. It follows that these solutions are-1 +- Solve x3 = 300x + 432 using Grard's technique, given that x = 18 one solution.

Explanation / Answer

let the other two roots be a and b. according to girards theorem : now a+b+18 = 0 (1) also 18a + 18b + ab = -300 (2) and a*b*18= 432 (3) ab=24 and from (1) a+b = -18 hence the equation becomes x^2 + 18x + 24 = 0 a,b= {-18 +- sqrt(228) }/2

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