First course in the theory of equations

by Leonard Eugene Dickson

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. 1922 ...be no other root between the same limits. But in the contrary case, narrower limits are necessary, such as 4 and 4.3, with the further fact that only one root is between these new limits. Then that root is said to be isolated. If an equation has a single positive root and a single negative root, the real roots are isolated, since there is a single root between--oo and 0, and a single one between 0 and + 00 However, for the practical purpose of their computation, we shall need narrower limits, sufficient to fix the first significant figure of each root, for example-40 and-30, or 20 and 30. We may isolate the real roots of f(x)=0 by means of the graph of y=f(x). But to obtain a reliable graph, we saw, in Chapter V that we must employ the bend points, whose abscissas occur among the roots f'(x)=0. Since the latter equation is of degree n--1 when f(x)=0 is of degree n, this method is usually impracticable when n exceeds 3. The method based on Rolle's theorem (§ 65) is open to the same objection. The most effective method is that due to Sturm (§ 68). We shall, however, begin with Descartes' rule of signs since it is so easily applied. Unfortunately it rarely tells us the exact number of real roots. s-67. Descartes' Rule of Signs. Two consecutive terms of a real polynomial or equation are said to present a variation of sign if their coefficients have unlike signs. By the variations of sign of a real polynomial or equation we mean all the variations presented by consecutive terms. Thus, in X6--2x3--4x2+3=0, the first two terms present a variation of sign, and likewise the last two terms. The number of variations of sign of the equation is two. Descartes' Rule. The number of positive reed roots of an equation with real coefficients is either equal to the ...

38 pages, Paperback

First published September 1, 2009

About the author

Leonard Eugene Dickson (January 22, 1874 – January 17, 1954) was an American mathematician. He was one of the first American researchers in abstract algebra, in particular the theory of finite fields and classical groups, and is also remembered for a three-volume history of number theory, History of the Theory of Numbers.

Reviews

[Aryan Prasad · Rating 2 out of 5]

Even though the book is labeled "First Course..", I found it too basic. Neither is the prose good, questions good nor the content is rigorous enough. Had to go through, because it was the recommended reading in my college.