On the Theory and Solution of Algebraical Equations; With the Recent Researches of Budan, Fourier, and Sturm, on the Separation of the Real and Imaginary Roots of Equations

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1835 Excerpt: ...lately, it was not in the power of algebra to furnish these means, except in a few instances, and the computist was in consequence often involved in doubt and perplexity, from being unable to separate the imaginary roots from the real. This separation had indeed long ago been shown to be theoretically possible, by the celebrated Lagrange, but the numerical labour, which the process involved, rendered it practically useless, except in the simplest and least doubtful cases; it is now, however, entirely superseded by the method of Sturm, the exposition of which will form the subject of the next chapter. For further researches on the subject of the present Chapter, the student is referred to Mr. Horner's series of papers, in vol. 5 of Leybourn's Repository. On the Determination of the Integral Roots by the Method of Divisors. (63.) It was demonstrated at (13) that no equation in which the coefficients of the first term is unity, and those of the other terms integers, can have a fractional root; so that the roots of every such equation can comprise only whole numbers, and interminable decimals. These latter we have shown above how to approximate to as closely as we please; and, although the same method will furnish us, figure by figure, with every integral root also, yet it is worth while to explain here a distinct process for the discovery and determination of every such root. The method we advert to was proposed by Newton, and is called the method of divisors. Let N + Ax+ A2x' + A3x3 + A4x +.... - = 0 be an equation of the nth degree, in which the coefficients are all whole numbers; and let a be an integral root of it, then we must have N + Aa + A, a2 + A3a' + A4-4 +.... o = 0 N.-.--=--A--A2 a--A, a3--A4o3--....---"-', N from which we ...